I think it's less of an over-complication issue and more of a resource issue. It is widely known that at the outbreak of the war in 1939 the military might of France alone could have by most accounts defeated the German army. Despite impressive displays of machinery and armour the German's had a small stockpile of equipment, and access to few natural resources, mines, and reserves. As part of the WWI Treaty of Versailles their army and navy was limited, and they were forbidden an air force and submarines. In addition to this they had to pay reparations, reducing their economic power and GDP. The country nearly went through a civil war with open gang and political party violence in the street, and the new government struggled to maintain order, control, and to start righting the ship. There was little emphasis placed on a national, unified, singular development of arms for use against other armies or countries. Other European countries didn't have these same restrictions, but suffered monetarily after the war. The UK famously tried to avoid another costly war by all means possible under PM Chamberlain, and did not exactly spend lavishly on advancing their arms. When the Nazis came to power in 1932 the great depression was reaching its end and Hitler almost immediately began a re-armament campaign and began growing the numbers of both the army and navy. Plus spending in public works like the Autobahn, money was starting to flow again, albeit slowly.

When the invasion of Poland happened in 1939 they armed forces were supplied on the back of 7 years work. The coal mining and industrial plants of Silesia in Poland were a boon, and the Nazi's invaded Norway for its coveted access to oil reserves. Beyond the humiliation of Stalin and communism, the vast oil fields of the USSR steppe were also highly sought after. As early as 1941 the German army was running out of oil, and IG Farben was contracted to develop a synthetic oil for them in 1942. With campaigns across all of Western Europe and USSR, the manufacturing capacity of Germany could not match the army's growing demand. Many of the coal mines in occupied countries had been sabotaged to varying degrees, delaying the delivery of essential minerals, and resources were being consumed to construct new ministries and grand buildings for the Nazi HQ's in Berlin, road and public works jobs, and high ranking Nazi members new lavish homes/estates. The lack of financial, natural, building, and human resources plagued the development and manufacturing of arms and war accessories from the beginning, much more than any perceived complexity in design. It just didn't all come to light until complications arose out in the field, supply lines got stretched, losses began piling up, and demand peaked/spiked.

Regarding Speer, his predecessor Fritz Todt was an engineer with zero grand scale manufacturing experience. He began his career overseeing water tower and road works projects, hardly of a scale near what the many divisions of the German armed forces would require years later. He was also the first person to have the job, so there was not exactly a "how to" or an existing structure in place for him to follow or adapt. The demand for army supplies resulted in a civil conscription for mandatory labour, which also meant many unskilled, new workers joined the labour force. Once the armies losses started to pile up, more and more workers were sent to the fronts, thus repeating the cycle of long learning curves. Speer initially only controlled arms for the army, when he got the navy in 1943 it was basically a disaster. He tried to have new submarine designs made via modular construction (different parts built at different shipyards then assembled later), but had almost no success. By this point the Atlantic was pretty safely in the hands of the Allies and the Kriegsmarine would cease to be a real threat. He seemed to have middling success with the Luftwaffe, and production did increase, but once again the superiority of the combined allied air forces vastly out powered and out numbered what Germany could produce with an ever weakening labour force. In addition to this, by 1943/44 Germany had essentially no oil, so even if he did improve things, they couldn't get the machines to run, and even more of a concern was getting people to pilot or drive them as it was becoming ever clear after their defeat in the USSR there was little chance of victory. His appreciation for grandiosity and design did result in the V2 rocket really rapidly advancing during 43/44, but again by this point it was of little consequence.