I stopped receiving Outlook for mac 2011 version 14.7.7 emails on 1/27/2021 on my macOS High Sierra version 10.13.6I

Not if you remove only the app itself. All of your emails, contacts and other Outlook data are buried away in your user account. Once you remove and reinstall Outlook, it will automatically pick up your email data since it always looks in that location for a Main Identity folder.

Ah! Okay, that makes sense. MS killed the activation/verification server for Office 2011.

What replacement you get depends on your needs.

1) Office 2019. Perpetual license software. Just like Office 2008, 2011, 2016. Bad news. MS changed how it supports the perpetual releases. Before, you could use it under as many Mac OS upgrades it would work in. For Office 2011, that was almost 8 years. Not anymore. Their new rule is such versions will be supported for only three major Mac OS releases. Office will simply refuse to run in the next release. Since 2019 came out with Mojave, that means the three OS versions Office 2019 will run in should be Mojave, Catalina and Big Sur.

MS said there will be at least one more perpetual license release. Whatever that will be named, expect it to be released just before, or alongside with whatever Apple calls the next major OS after Big Sur. That means what you spend on Office 2019 now will only be good for one year if you always keep the OS current. Then you'll have to pay full price again for the next perpetual Office release.

2) Office 365. There's more to it than just monthly fees (or once a year). 365 always gets the most current features when they're ready. Office 2019 is feature stuck where it was at its release. For a single user, the perpetual license (if you get the full three years worth out of it) is still cheaper than even the $70 single license of Office 365. But as soon as you have to computers to run 365 on, the $100 per year option is less than two individual perpetual licenses. And a LOT cheaper if you use all six.

Hmm, there's conflicting info in your post.

First and foremost, you say you can access all of your email with Apple's Mail app. That alone says there's nothing wrong the various email servers you use (personal, former employer, university). That means something is specifically wrong with Outlook 2011.

MS started using a login verification of software ownership with Office 2016. That continues on into Office 2019 and with Office 365. Office 2011 and older have no such "Microsoft Account number". You install the app, enter the product key included in the box (or emailed to you if purchased as a digital download), and it works. There is no other connection, other than MS's servers checking the validity of the product key to complete the activation.

Anyway, something is wrong with the account info in Outlook. And I mean email account, not MS account. Open Outlook. From the menu bar, choose Tools > Accounts. Check to make sure all of the fields are identical to the one's in Mail (incoming and outgoing server, port, email address, etc.).