Yes, it's more-or-less the case that you need the current Rust compiler if you want to compile a bunch of Rust projects and have no issues. But there's really no downside to this. Just grab the latest Rust version. It's free and has no breaking changes.
The only time I can see it being an issue is if you're trying to use the Debian/Ubuntu version of rustc or cargo. The version of rustc on Debian stable is 1.48, which was released in Nov 2020. I'm not sure what advantage there is to shipping such an old version of rustc when it guarantees backward compatibility, but if this is what you have then, sure, a big chunk of the Rust ecosystem won't compile.
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I still think your original comment was misleading because you said the Rust mentality was "move fast, break things". This almost always refers to a lack of backward compatibility, when existing software no longer works with newer versions of the thing that changed.
The Rust mentality is to use all new features immediately without consideration. And this is entirely on the Rust devs which are, naturally, for now, the bleeding edge types.
Misleading? No. This is the reality of most Rust dev behavior. But you're right in spirit, the problems aren't intrinsic to the Rust language itself.