This is a myth perpetuated by selective cultural memory. The Nazis were all about "free speech" before they were in power. They frequently complained about "Free speech" when papers published editorials criticizing them, or when their public appearances were protested, or when opinion pages didn't present "both sides".
One widely used Nazi propaganda poster from 1928 even showed Hitler with a big "CENSORED" block over his mouth, captioned "Only One of the 2000 million people in the world is not allowed to speak in Germany"[1]
> in speech-tolerant societies, a diversity of ideas compete and the best, most valid ideas rise to the top irrespective of bias.
I'm not sure that you understand what "bias" means. Bias is that which leads people to believe in ideas that are false for various sometimes difficult to quantify reasons. The fact that evidence opposing flat-eartherism is available has not prevented that idea from gaining popularity, in fact it is more popular today in 2021 than it ever has been.
The "marketplace" metaphor itself exposes that the argument comes from a utopian perspective, the argument only works if you pre-suppose that a free and unregulated economic market produces best results, but we know from experience that it produces monopolies, child labor, and bread doped with sawdust.
Public discourse is like a market in some ways, namely that people with more money are able to advance their ideas further by buying media outlets and marketing broadly and engaging in deceptive communication that exploits cognitive bias to acquire more believers, in just the same way as people with more money are able to buy out and undercut competition and exploit addictions and consumer psychology to get more sales than their products deserve.
1. https://twitter.com/_amroali/status/1190809322315534336?s=20
The first thing nazis did was to revoke free speech rights. That's an example of authoritarianism and free speech being opposites.
> Indeed, authoritarianism in general and Nazism, Stalinism, Maoism, etc in particular always starts with speech restrictions
Authoritarianism virtually requires restrictions on speech to come into its own.
Moreover, "marketplace of ideas" is a metaphor for what happens in a society that has a high degree of free speech: the best most valid ideas rise to the top. You argue that the marketplace metaphor doesn't work because bias exists and because ideas vary in validity, but that doesn't make sense--these facts are the very mechanism by which the marketplace metaphor works: in speech-tolerant societies, a diversity of ideas compete and the best, most valid ideas rise to the top irrespective of bias.