1. Use labels/filters to tag the emails from the ones you want to keep.
2. Use the filter to Archive the emails with those labels.
3. Everything else remaining in the inbox can be assigned to a new label called "deleteme"
4. archive everything with that label

At this point there should be nothing in the inbox. Of course all the emails still exist, the archive function just removes inbox label.

If you are running out of space, then go to the label "deleteme", select all the messages, and click the delete button.

Based on my addition to the initial question, I feel like the answer is no (sort of).

The emails in question were skipping the inbox due to a "hidden" filter in the import settings for that address. This filter was not visible in the Filters section of the settings, and it was not applying a label, so there was no "easy" way to track down what was causing the problem.

The easiest way to track down what filters get applied is simply to make sure that every single filter (even these "hidden" ones) applies a label. That way, the label will quickly tell you what filter gets applied.

That's an easy solution for filters created under settings, but not quite so easy for the filters tucked inside Gmails import settings for other addresses.