There is always more to say on any topic, but one of the leading theories for the voters becoming more sympathetic to the Nazis and building on their base is the so called "radicalization of the middle class" which was first theorized by Seymour Martin Lipset in the 1960s and expanded upon as a theory for the rise of the Nazis.

Some background. You mention Versailles and the Great Depression in your post, but it wasn’t really Versailles or the Depression that radicalized people. Or, at least, a long time passed between Versailles and the Depression. However, that’s not to say that the Treaty of Versailles, or rather, the Government response to the treaty, didn’t have an effect. Larry E. Jones paints the picture that the government response to Versailles, namely the self-inflicted hyperinflation, vastly redistributed wealth among all facets of German society. One major contributor to the radicalization of the middle class was small-business owners being driven out of business. Government price controls and lack of reliable access to goods caused many small businesses to close down. In this same time big department stores began consolidating and buying out competitors exacerbating the decline of small business owners.

For white collar workers it was just as bad. Salaried white-collar workers found their salaries quickly unable to keep up with the prices. This led to a flight from salaried positions to wage labor. Pensioners were also being forced back into the labor market for the same reason which caused mass unemployment amongst these people. Making things worse is that these white-collar workers often supplemented their pay with dividends from investments and savings accounts. With the hyperinflation these alternate sources of income also dried up.

To make matters worse Germany was already institutionally right-wing. College professors, bureaucrats, and the judicial system had a right-wing nationalist bias. This exacerbated the radicalization because it meant most white-collar positions already had a right-wing institutional bias. This bias shows the best when Adolf Hitler gets little more than a slap on the wrist when his Beer Hall Putsch fails in 1924. The judge presiding over that case was sympathetic to Hitler. It also meant everyone that got a college degree or worked in bureaucracy was already primed to be sympathetic to right-wing sentiments. The right-wing controlling the judicial system also gave them a distinct edge when Communist paramilitaries would clash with Nationalist and Fascist Paramilitaries. The left-wing agitators would get harsh sentences while the right-wing agitators would often get slaps on the wrist and sent back out.

Lastly, a large factor in German society prior to the Weimar Republic was the military, which also had a right-wing bias of course. The military presented middle-class youths the opportunity for social mobility. It also meant most older middle-class individuals had served in the military and got access to resources because of that. The disarmament of the military locked the new voting generation out of those resources their parents had. It also meant that a large number of right-wing veterans and their children were available to join paramilitaries. These paramilitaries were considered respectable by the middle class and so joining one was socially beneficial which bolstered right-wing and NSDAP ranks. It also gave the Nazis a further advantage over Communist paramilitaries who were already disadvantaged structurally.

With that background out of the way, you asked in your post why people believed the Nazis would make things better where the other parties wouldn’t. Well following the hyperinflation fiasco, the Weimar Republic was governed by the so-called “Grand Coalition” more-or-less run by Gustav Stresemann. He was chancellor for a short period of time in 1923 and then took a back seat role in the government as foreign minister until his death in 1929. Stresemann united the major political parties of Germany the SPD (social democrats), the Centre Party (Catholic centrists), the DVP (right-wing liberals), and DDP (left-wing liberals). During his time as foreign minister he successfully ended hyperinflation by negotiating the Dawes Plan (and later Young Plan, but I’ll refer to this scheme as the Dawes Plan from here on out as the origin was the same and the Young Plan collapsed before it got started) which reorganized the banking system, got Germany on track to repaying war reparations, and, importantly, backed the German economy with US Banks.

His death in 1929 coincided with the beginning of the Great Depression, and remember that the Grand Coalition backed the Dawes Plan as a way to fix hyperinflation? The Dawes Plan was backed by US Banks which meant that when the US economy collapsed the Germany economy no longer had it’s chief backer and it too collapsed.

Following this development the government then pursued a number of failed policies that only served to exacerbate the depression, namely, austerity measures aimed at causing deflation and lowering government spending. This left a lot of people deep in debt and had the government cutting expensive programs and raising taxes.

The Nazis then took advantage of all of these factors to form a successful political party. The Nazi political base was middle class, rural, and/or white-collar protestants. The Nazis pointed to the hyperinflation and Dawes Act as redistributing wealth to the wealthy elite aka Jews. The Nazis blamed the hyperinflation on the Treaty of Versailles. The Nazis further argued that the hyperinflation was done on purpose to strengthen Jewish-Bolshevik elites. The Nazis argued that Jews owned the big department stores and the banks and that the Jews directly benefitted and controlled the Grand Coalition, which, if you’ll remember, included the conservative parties. Furthermore, the conservatives were behind the unpopular austerity response to the Great Depression which meant their support waned, and when their support waned it played directly into the Nazis who were seen as the alternative.

Economically Nazi political campaigners were more or less allowed to just say whatever they wanted to get the vote out. The Nazi ideology was more focused on racial and nationalist things and so economically they could promise whatever the locality they were campaigning at wanted. If the voter base was historically white collar then the Nazis would bend their message to them, if the voter base were so-called “Tory Workers” or white-collar people forced into blue collar work the Nazis would completely change their message to appeal to them, if the voter base was farmers the Nazis would change their economic message to fit them etc. This mean that as long as you bought the argument that the pro-republican parties were controlled by the Jews and that Versailles and it’s consequences caused the problems in your life you had a place within the NSDAP because they would promise you whatever economic policy you desired. The Nazis win 18% of the seats in the 1930 election making it impossible for Pro-Republicans to form a coalition. This gives Hitler the opening to take power, which is outside of the scope of this question.

There is always more to say, this is only one theory (and quite an old one so there may be newer research that shows different things), and this is an extremely complex topic. However, I hope this gives you a better understanding of the political situation at the time and why the people who voted Nazi switched from the pro-republican parties to the Nazi party. I also hope you see how people voting for the Nazis wasn’t directly linked to the Treaty of Versailles or Great Depression, but that the unique circumstances in Germany meant that the Nazi argument, as flimsy as it was, still had some basis in fact and it meant that if you were already primed for that argument, as many Germans were, you could come to the conclusion millions of Germans came to and vote Nazi.

Sources:

Political Extremism in the 1920s and 1930s: Do German Lessons Generalize? The Journal of Economic History , Volume 73 , Issue 2 , June 2013 , pp. 371 – 406 DOI: https://doi.org/10.1017/S0022050713000302

Jurgen Falter, Radicalization of the middle classes or mobilization of the unpolitical?: The theories of Seymour M. Lipset and Reinhard Bendix on the electoral support of the NSDAP in the light of recent research Volume 20, Issue 2 https://doi.org/10.1177/053901848102000207

Larry Eugene Jones, Inflation, Revaluation, and the Crisis of Middle-Class Politics: A Study in the Dissolution of the German Party System, 1923-28, Central European History Vol. 12, No. 2 (Jun., 1979), pp. 143-168

Dick Geary, Who voted for the Nazis? (electoral history of the National Socialist German Workers Party), History Today, October 1998.

Richard F. Hamilton, Hitler's Electoral Support: Recent Findings and Theoretical Implications, The Canadian Journal of Sociology, Vol. 11, No. 1 (Spring, 1986), pp. 1-34

I'd like to ask a follow-up question, if I could. This is directed primarily to u/thewildshrimp, but it's for anyone who would like to answer it.

So, the middle- and working-class Germans in u/thewildshrimp's description accept the Nazi narrative of Germany's economic collapse. This pins responsibility on big businesses, e.g., department store owners and bankers, many of whom are Jews. Thus, in the Nazi narrative, the Jews, the traditional outgroup, become enemies of the Volk and engineer the German economic collapse.

However, to my knowledge, department store owners and the like would not in fact have been at the top of the pre-War economic pyramid. The top layer was occupied by the heads of large industrial concerns, and (again to my knowledge),they were almost all Gentiles. (Emil Rathenau of AEG is one of the few exceptions I can think of.) Next to them, department store owners had only mediated and middling economic power.

On reflection, the middle and working classes could have understood by this that the Nazi narrative was false, because the people Nazis blamed for Germany's collapse did not have enough power to cause it -- except that they would have had next to no contact or familiarity with the real power in the circles of top-level industrialists, . On the other hand, department stores and local banks were known and powerful enough in their localities, visible and available as targets for revenge by the baffled and frustrated ranks of the newly impoverished. Anger could be vented on them safely.

I am from the US Great Lakes area, the so-called Rust Belt, so there is a good bit of presentism lurking in my remarks, I suppose. I think I see some very similar dynamics in the US: white middle and working class people being encouraged to target "affirmative action/ wokeism" or "illegal immigrants" as responsible for their own job and pension losses, simply because these targets are visible and available for blame.

I'd be glad for your response, and thank you.