I GOT AN ANSWER!!!!!!
1) Yes, the new Passwords app is SUPPOSED TO AND EXPECTED TO store and update passwords. Wheeeew! I got admonished already for having made that assumption. Now a senior advisor lady just told me, "Of course! That's the whole point of the app!" OK, now we're getting somewhere.
2) The first tier of advisors didn't know what I was talking about, until one of them said this morning, "Oh no, you have to WRITE YOUR PASSWORDS DOWN IN NOTES OR ON PAPER SO YOU REMEMBER THEM." I asked, "OK, so then all of this is just suddenly completely different from before two weeks ago, yet Apple didn't bother to tell us that our Apple ID passwords can no longer be stored? Your advice is 100% contradicting everything apple has told me for the past 4 or 5 years. Wouldn't they make a point of telling us all about this drastic change, where we go back to 1980's security practices and write our passwords down, maybe using post-its on our monitor? Wouldn't they make sure we all knew about that kinda change, since we have to be logged into our apple stuff at all times to use anything?"
Turns out, my mac required (but didn't prompt me) a restart for the changes to my iCloud password to take effect for some reason. Great. It also turns out that the SOLUTION was to manually log into iCloud online, then change the password, then a very familiar prompt pops up asking if you want to store your new password in Passwords.
I'm appreciative that my issue is resolved, but man... Apple probably still has the best customer support, but not from the first tier of folks who answer the phone. I don't know what or when it changed, but the first tier seems like they might be working support for several companies at once, just looking at prompts on the screen. They often seem unfamiliar with Apple products.