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Go is self-hosting (i.e. Google's Go compiler is written in Go and is able to compile itself). I would assume that includes the garbage collector.

Nope it does not. Writing the compiler self hosting has nothing to do with writing the run time system.

There's some assembly in there, but it seems to be mostly Go.

People have written GCs for lisps in (non-allocating subset of) the target language

Yes they have. It's a great approach. Does Go have a precisely defined subset for this?

Yes, in fact that is the case with the reference compiler.

You need a Go compiler to compile Go, so yes.