Answer: it's in the Contacts, but somewhat hidden.
For people you have in a Contacts app, gmail gives you a top level "My Contacts". Click the dropdown next to the word Gmail in the upper left of the standard Web UI and select Contacts. Editing or deleting things here is a little tricky but not bad.
There is also a "secret" stash of "approximately everyone you ever mailed to or replied to EVAR" and that's the source of a lot of old/stale/bad address completions. Here's how to find that.
While in Contacts, click the dropdown next to "My Contacts". You might need to push Chat/Hangouts farther down the column to make space to see this. Below My Contacts are other frequently used contacts, any "Starred" contacts, any email groups you've created, any imported contacts, etc.
Below that are your G+ circles. Below that is "Most Contacted". And below THAT is a thing called "Other Contacts".
If you're like me, there are hundred or thousands of addresses in there, many of them old/bad/deprecated/should not be used. Find and delete them.
(Nearly every Web page I read on this topic was wrong, or out of date in giving this advice. Probably there are relevant pages somewhere that tell you all this, but I couldn't find them.)