The title isn't quite right, though. Boxing, method lookup, etc. come under "interpreter" too.
There's a continuum of implementation between a naive interpreter and a full-blown JIT compiler, rather than a binary distinction. All interpreters beyond line-oriented things like shells convert source code into a different format more suitable for execution. When that format is in memory, it can be processed to different degrees to achieve different levels of performance.
For example, if looking up "str" is an issue, an interpreter could cache the last function it got for "str" conditional on some invalidation token (e.g. "global_identifier_change_count"), so it doesn't in fact need to look it up every time.
Boxing can be eliminated by using type-specific representations of the operations, and choosing different code paths depending on checking the actual type. Hoisting those type checks outside of loops is then a big win. Add inlining, and the hoisting logic can see more loops to move things out of. Inlining also gets rid of your argument passing overhead.
None of this requires targeting machine code directly, you can optimize interpreter code, and in fact that's what the back end of retargetable optimizers looks like - intermediate representation is an interpretable format.
Of course things get super-complex super-quickly, but that's the trade-off.