This would place a burden on all iterators to implement a large and complex interface. This goes directly against the philosophy of Python protocols, which is to be as minimal as possible. Do one thing, and do it well. Agreed.
And where do you stop? You've picked an arbitrary subset of things one might want to do with an iterator. Why those particular ones? I've picked them arbitrarily for the example.
What about the contents of the itertools module? Should they be included too? Why or why not? I imagine it to include those from itertools module too. And those from builtins like zip and enumerate. If you ask me why, I guess one way I reason about this is that these are what I think an iterator should be able to do (at the expense of a bloated class).