FDR did not pay the farmers as a way to get their vote. He already had their vote. Furthermore, allowing prices to fall would have done nothing for many of the victims of the Great Recession, as many of them did not have jobs and thus could not afford to pay for food no matter what it cost.
The Federal Surplus Relief Corporation was created for the sole purpose of maintaining America's agricultural output, which then and now, was the primary source of America's economic strength. Maintaining food production also meant that many of the associated farming jobs would remain intact, preventing the crisis for worsening.
The farming jobs supported many skilled industrial production jobs, especially for heavy equipment such as tractors. This would prove invaluable a decade later when the U.S. entered WWII and the factories began producing weapons and tanks. The factories already had plenty of trained workers ready to work the lines.
All of this is covered in most high school U.S. history courses.
That's partially correct. FDR wasn't just trying to buy farmers votes, he really did believe that high prices were good for the economy. He correctly realized that the deflation being experienced in the US was bad, but he thought that artificially increasing prices would fix things, but in reality people didn't have the money to pay the new, higher, prices and all of FDR's efforts to raise the prices of things (the NRA especially) just ended up making the depression worse.
Now, FDR did a lot of things and some of them worked pretty well. He'd touched off the fastest industrial expansion in US history a few months before he killed it with the NRA, for example. I'm not aware of any current economic school of thought that would endorse the idea that price supports actually helped with the depression. A Keynesian would say that you have to run a deficit to increase the aggregate demand, a Monetarist would say you need more money to increase aggregate demand, a Supply-Sider would say you can't help, a Socialist would say the state needs to take over the means of production, etc.
Now, I do have friends from states that rely on agricultural subsidies, and their high school history textbooks evidently did wax poetic about how awesome agricultural subsidies were, but that isn't in most American's high school educations.
Arms and weapons alone (one of the main perceived sources of "America's economic strength" abroad, along with intellectual property and others) make more than that (sources estimate between 1% and 4% of the US GDP).