A general purpose compiled language should be suitable for writing its own compiler, but I don’t think the same holds true for interpreted languages.

A language can be suitable for writing a compiler, but if there is another language that's 10x faster that's also suitable, then you're losing out on a lot of compilation speed for no reason.

Dog-fooding a language by writing a compiler in it can lead to the designers adding language features to make compiler development easier, even if they detract from the design of the language for the 99% of users who aren't writing a compiler.