Move some HandlerInner functions to Handler.

`Handler` is a wrapper around `HanderInner`. Some functions on
on `Handler` just forward to the samed-named functions on
`HandlerInner`.

This commit removes as many of those as possible, implementing functions
on `Handler` where possible, to avoid the boilerplate required for
forwarding. The commit is moderately large but it's very mechanical.

Author: nnethercote