No, this is false. For Rust codebases that aren't doing high-peformance data structures, C interop, or bare-metal stuff, it's typical to write no unsafe code at all. I'm not sure who told you otherwise, but they have no idea what they're talking about.
Our experiences are different.
Good developers only write unsafe rust when there is good reason to. There are a lot of bad developers that add unsafe anytime they don't understand a Rust error, and then don't take it out when that doesn't fix the problem (hopefully just a minority, but I've seen it).
It's the classic "misunderstanding" that UB or buggy unsafe code could in theory corrupt any part of your running application (which is technically true), and interpreting this to mean that any codebase with at least one instance of UB / buggy unsafe code (which is ~100% of codebases) is safety-wise equivalent to a codebase with zero safety check - as all the safety checks are obviously complete lies and therefore pointless time-wasters.
Which obviously isn't how it works in practice, just like how C doesn't delete all the files on your computer when your program contains any form of signed integer overflow, even though it technically could as that is totally allowed according to the language spec.