Your question in general is difficult to answer definitively considering the speculative nature of the counterfactual. Instead of answering in a simple yes/no fashion, I will attempt to provide some context for what was going on over the period of the Nazi regime with respect to the great powers and the areas in which they had major interests.

Hitler and the NSDAP were the driving force of the conflict as we know it, however there were ongoing conflicts that were influenced by the major actors in the second world war, or that merged with WWII. Additionally, there were unanswered questions, territorial disputes and general powder kegs in Europe or the territory of European powers, and many of these were ignited by the conflict or resolved as a part of it.

• The Second Sino-Japanese War

The traditional start-date for World War II is September 1, 1939, however there is an argument that can be made that the Second Sino-Japanese War was the "start", since Japan eventually found itself at war with the Western Allies, and their invasion of China was a major factor in the breakdown of relations between the United States and Japan.

• The Spanish Civil War

The Nationalist Spanish faction in the SCW received extensive material and manpower support from Italy, Portugal and Germany. The Republican faction received support from the USSR primarily, and some support from Poland and Mexico although they were generally required to pay for the equipment they received.

• The First and Second Vienna Awards

With the breakup of the Austro-Hungarian empire, Hungary had lost an extensive range of territories, and many ethnic Hungarians existed in what was at the time part of Romania and Czechoslovakia. The Germans and Italians arbitrated these territorial disputes in 1938 and 1940, granting parts of Slovakia and the region of Northern Transylvania to the Hungarians.

• The issue of German rearmament and the long term viability of Versailles

Under the Weimar republic, the German state had begun rearmament on a small scale, and had been violating the treaty of Versailles by retaining the General Staff (in an under the table way), by allowing the army to be larger than its paper strength indicated, by allowing arms manufacture to take place in other countries (i.e. Sweden with a connection between Bofors and Krupp, or the construction of aircraft factories in the USSR) and by collaborating with foreign countries to develop new designs of things they weren't supposed to have at all (i.e. Finnish-German cooperation on submarines). Rearmament started on a much larger scale under the Nazis, however it is questionable as to how long the Weimar republic would have allowed itself to be shackled by it, even in the absence of a militant nationalist government.

• Indian independence

Indian independence movements had been gaining steam throughout the 20th century, and by 1937 India had obtained a limited degree of self-rule. During World War II, an alternate Indian government with an army was established with the support of the Japanese, which hoped to inspire rebellion against the British. It did not succeed in this, nor as a fighting force in its own right, however civil disobedience did occur and the demands for full self rule did grow.

• Mussolini's expansionist desires

Italy had designs on Albania over most of the inter-war period. This culminated in the invasion and annexation of Albania in 1939. Italy had territorial disputes over several parts of Yugoslavia. Italian nationalists had irredentist claims that dated back centuries including Corsica, Corfu and Nice. In 1935-36, the invasion of Ethiopia saw the country become a colony of Italy. Under Mussolini, ideas about Italian imperialism were fostered in various corners of the country, and both direct action by Mussolini himself and later the broader global conflicts that waged allowed Italy to pursue its expansionist policies. Designs on Greece would traditionally be blocked by the United Kingdom. France was more than capable of fending off Italian incursions into its territory. However, with the fall of France and the (seemingly) weak position of the British, the Italians were in a position where it seemed possible to pursue all of the above goals.

• Stalin's opportunism and long term ambitions

The Polish-Soviet war in the early interwar period was not resolved to the USSR's satisfaction. The Poles claimed additional territories in what is now Belarus and Western Ukraine. Additionally, the Baltic States and Finland had gained independence in the aftermath of World War I. The USSR spent the 1920's and 1930's developing economically, and Stalin dedicated considerable effort to building up militarily in the latter part of this development. Stalin initially took measures to counteract fascist influence, however when alliances with the Western Allies and the Poles seemed to be hitting road-blocks, the USSR instead sought to temporarily ally with the German Reich. Stalin's foreign policy in this period can be seen as very opportunistic, taking advantage of the distractions provided by both the Japanese and the Germans+Italians in their respective conflicts to annex Eastern Poland, annex the Baltic States, wage a war on Finland and demand territorial concessions from Romania. It is difficult to say which, if any of these goals would have been pursued in the absence of the Molotov-Ribbentrop pact, however if a golden opportunity presented itself, Stalin was not above making aggressive moves.

---

Without Hitler's influence, it's likely that many of these things would have been reduced in scale or perhaps not happened at all. However, underlying tensions did exist, and there remained a possibility for a variety of conflicts or diplomatic incidents to occur in relation to them. Whether any of these could have acted as a trigger for a full scale world war is ambiguous. If a world war did break out without Germany at its centre, it would not have resembled WWII as we know it today, nor would it necessarily have broken out in the same time period.

There is an additional question that is related to whether other Nazis in power would have started World War II if, for instance, Hitler had been assassinated in the 1930's. In The Wages of Destruction, Adam Tooze presents an argument that the buildup of the German military necessitated the subsequent conquests in order to survive economically - that is, they were running a war economy with the explicit intent of being at war in the near future. Thus from a purely economic viewpoint, one might say that the outbreak of war was in some sense inevitable, however that seems to be a little bit too abstract of an argument to make. I would agree though, that those in power were absolutely intending to push for a war against the Czechs initially, then later the Poles, although they would certainly have been happy for the Western Allies to sit those conflicts out. Thus, the question of whether or not WWII would break out may be a matter of whether Hitler's replacement would blink in the face of Franco-British guarantees, rather than whether or not they had expansionist desires. Opposition to war from within the Nazis state was not always coming from a desire for peace as much as it was coming out of a fear that the German state may not be able to win such a war, and thus be repeating the humiliation of the end of WWI.

Whether other Nazi officials would have followed a similar path to Hitler would presumably depend on whom, specifically, ended up with the reigns of power, however the individual personalities of these figures are not known well to me. I will therefore leave that to others to discuss.

I refuse to speculate on counterfactual history re: Hitler never being born or something along these lines. But looking at the German political landscape of the 1920s and 30s, the Nazi regime's preparations for war and their revanchist agenda was a victory for certain elements in society and political power who supported a war or were willing to go along with it.

Those longing for a second war in Europe in Germany were during the Weimar Republic confined to a certain strand of extreme right political thought and certain swaths of the military but were never in a majority position. We tend to view Weimar as this republic doomed to fail but that is a historical assessment that doesn't fully hold up. The German political and even military leadership during the 1920s had made headway in circumventing the Versailles treaty with regard to the German military by outsourcing certain parts of their training to a joint program with the Soviet Union as well as re-negotiating certain terms of the treaty with regards to reparations. Generally, the memory of the last war, its hardships and problems were still rather present within the minds of a considerable swath of German society.

The Nazis as a political movement and other similar extreme right-wing movements drew in a lot of the people who had not experienced the First World War first hand but had nonetheless lived through it. Michael Wildt describes them in his book about the later leadership of the Reich Security Main Office. Born between 1900 and 1910, they were part of the "Kriegsjugendgeneration", the generation too young to fight in WWI but consciously living through it. Influenced by the image presented of the war in propaganda and by the supposed honor and heroics of war, they formed a political opinion that saw compromise as failure and political victory only achievable by the physical annihilation of the political enemy. Going on to be students in the Weimar Republic, they formed a sort of ideological backbone of the Nazi movement. Even Heinrich Himmler can to a certain extent be counted among their ranks, being born in 1900 and having never fought on the front line, which he regarded as a personal failure.

With the Nazis coming to power, their political program was aimed towards war and they worked hard to remove those who could have politically stood in their way from positions of power. From the more moderate right wingers removed during the Night of the Long Knives to the Blomberg-Fritsche affair four years later, influential people within the political system and the military were removed in favor of those who went along with starting a new war or were heavily in favor of it.

It is important to keep in mind here that the decision to wage war was not made in 1939 but rather in 1936. Standing at a crossroads of how to organize their economic system wrt foreign exchange and the import of raw materials, the so-called four year plan was the decision that Germany was to wage war within the next four years. It explicitly says so in the memorandum Hitler himself wrote. However, that would not have been possible if the leaders of the big companies and the military would have refused to go along with it. And go along they did.

The army and its leadership in form of people like Ludwig Beck were enthusiastic about the venture, because it meant enlarging themselves, reintroducing conscription etc. etc. Similarly, the industrialists, many of whom saw a massive chance for profit in the war. Especially Krupp but also the IG Farben leadership wanted the investments necessary for rearmament programs.

In terms of the German public, the war in its beginning was not viewed overly enthusiastic. It is always hard to gauge public opinion in a dictatorship like the Nazi regime but the invasion of Poland was as far as can be told viewed with a certain apprehension. It was the victory over France which restored a lot of confidence and enthusiasm for the Nazis' war.

In light of all that, a war waged by Germany without the Nazis is as far as can bet old less likely or at least the war that they in the end fought is less likely. While there were elements within the German political and military leadership willing to go to war until the Nazi regime took over, they were not in a majority position. Their support was to become essential under a regime hellbent on going to war but before such a regime was in place, a war agenda didn't have the political pull to be realized. How this would have changed with a different right wing regime in place that was also interested in fighting a war is difficult to say but the answer to your question as best as possible is that while there were others in power who wanted a war, they needed the Nazi regime in order for their agenda to become reality.

Edit:

I just saw the edit you made to your description, OP. I can not overemphasize how much of a problem I have with reducing the Nazi dictatorship to the person of Adolf Hitler. Hitler, while being an important factor, relied on a whole slew of people and social forces to support the political agenda of the Nazis and was far from the only one setting that agenda. As I tried to point out above, an important backbone of the Nazi movement were people who saw war as noble and necessary. They did not represent the totality of German society but without them as well as without large parts of the military leadership and the industrialists of German on deck, war would not have been possible.

Sources:

• Michael Wildt: An Uncompromising Generation.

• Adam Tooze: The Wages of Destruction.

• Richard Evan's Third Reich trilogy.

Did the SPD or KPD have any interest in defying the Versailles accords? If so, to what extent?