The existing implementation stops recording type information once it
encounters an error. This results in missing type information that is
needed by various tools. This change handles a few commonly encountered
cases by continuing to check subtrees after errors. Also, add tests for
cases where the package fails to type-check.

Updates #22467

Change-Id: I1bb48d4cb8ae5548dca63bdd785ea2f69329e92b
Reviewed-on: https://go-review.googlesource.com/123578
Run-TryBot: Rebecca Stambler <rstambler@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Alan Donovan <adonovan@google.com>