Your interpretation is correct, but so-called Internet memes are not overly concerned about grammaticality or idiomacity or logic.
The humor fails on multiple levels. As Edwin and Dan point out in the comments, there are various practical objections, for example, that toasters toast bread. But it is an attempt to be humorous, after all, not really a political argument, so the bad analogy is beside the point.
The attempt lies primarily within the thrice-repeated toast. This forces the reader to pause to work out how each toast functions. Toast is not the grammatical verb form, however, which spoils the attempt. Guns and people are plural whereas the middle toast, being used as an uncountable noun here, is treated as singular. The grammatical form doesn't succeed as there is no confusion as to which words serve which roles in toast toasts toast as compared, for example, to Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo.
You need a noun which shares a form with a related verb and which can also serve as its object or other complement. Consider
Fishermen don't fish fish, fish fish fish.