This was rather disheartening.

I've always treasured the diligence of the anti-cheating monitoring here; it's what makes playing here worth it.  You take for granted that it works magically and effectively until it turns an eye on you:

Of course, you can't take anyone's word for it, but not once in my entire 12-year membership have I cheated.  I don't even feel like I'm playing particularly well, and I am not strong at tactics.  I haven't been winning at an unusual rate lately -- just as many blunders as always.  I don't know what the algorithm is even looking at.  Here's my rating over "all time":

I've gotten no better (in fact, a bit worse) over 12 years... and yet here I am, wondering what I should be doing.  Not getting to play here would itself be pretty disappointing, but more than that, I consider my reputation a professional character asset.  I have no desire to have my account closed for "Fair Play" violations.  Do I just need to play out my games -- and not too well! -- and give up the site?

I don't expect an answer from the admins.  I'm sure you can't give one anyway due to the proprietary nature of the fraud controls.  But I guess you can't be a decent player without running the risk of false accusation.  It just sucks.

Its not personal.  Whenever someone crosses 2000 they get that message.

Yep, it's just a warning for going over 2000. 

Since when?  I’ve been floating below and above 2000 for many years, and I’ve never received that message before.

I believe there's a message when someone first crosses 2000, but after that there's several reasons why a single game might be flagged as "potentially suspicious", hence this reminder; it sounds like you have absolutely nothing to worry about, so just carry on as you always have. Accounts only get closed for Fair Play violations when Chess.com is absolutely sure that an engine is involved, to the extent that they are willing to go to court over it.

For some insights into how Chess.com approaches this, check out some of the following links:
https://www.chess.com/blog/News/new-tools-for-fair-play-on-chess-com
https://www.chess.com/article/view/chess-com-fair-play-and-cheat-detection
https://support.chess.com/article/648-what-do-i-need-to-know-about-fair-play-on-chess-com
https://www.chess.com/blog/DanielRensch/cheating-on-chesscom
http://pubs.royle.com/publication/?i=469188&article_id=2985800&view=articleBrowser&ver=html5

Cheers,

David, moderator

Administration: I love playing on Chess.com, specifically because of your Analysis feature. Over the years the Analysis feature has suggested several tactics that I have incorporated into my play. Just today, it disappeared! I do not get recommended moves and wonder why I would continue to use this (used to be free) site. Please be informed that without the Analysis feature I used to see, you will lose someone to send your advertisement to. Thank you.

Are you using the mobile apps or running it on a web browser? Chess.com have been doing a lot of work on their Analysis features, and as they mention in their March 2020 update, they're bringing cloud based analysis to their mobile apps ie. the Analysis isn't going to run on your smartphone, but in the cloud, presumably giving faster and better analysis. I don't know if that means they've taken out the local Analysis while they get that cloud based on up and running, or if it's been done but it isn't compatible with older versions of iOS or something else, but if you've still got a problem after the next few days, let them know via Help --> Report a Bug, or contact the Support team directly via https://support.chess.com/article/346-contact-us