Edge removed all my programs, leaving desktop icons unusable

Welcome to tenforums. When you wrote your post, did you notice the forum prompted you for your full build number (so we don't have to keep asking). Thanks.

Please post a screenshot of your Update History (Settings) to show the update, thus:

It also removed all the system restore points.

Guessing, you have had a feature update to build 2004. Feature updates always remove restore points (and often disable system restore). Exactly as they should- you wouldn't want bits of the old build restored into the new one, would you.

System Restore cannot reverse a feature update anyway- there is a separate mechanism for that: 'Go back to the previous build'. (Tutorial available in the Tutorials section).

To be sure what to advise we need to know which update you received.

If you have received a feature update to 2004, you could use that.
Go Back to the Previous Version of Windows in Windows 10
*** That's assuming you've had a feature update, which I don't yet know, so this is an assumption at this point given that you haven't specified the update exactly.

If you received this update KB4559309, note this comment:
Microsoft confirms Windows 10 slow boot issue due to KB4559309 - MSPoweruser

Otherwise, it's possible you've been logged in with a temporary user profile.

Hi, you've not posted a screenshot of your update history or given an update number.

KB4559309 Update for new Microsoft Edge for Windows 10 - May 27
is the new Edge update, and I mentioned this previously:
Microsoft confirms Windows 10 slow boot issue due to KB4559309 - MSPoweruser

mentioning associated problems.

Note it also says:
In June Microsoft rolled out cumulative update KB4559309, which brought the new Chromium-powered Edge browser to Windows 10 users, replacing the old EdgeHTML Edge.
Unfortunately, after delivering a faster browser, the update also appeared to slow down Windows 10, with users complaining of performance issues, with slow boot times, slow loading of apps and ever web pages loading slowly.
Now, on Microsoft Answers, a moderator has finally confirmed that Microsoft Edge team is currently in the process of investigating this problem.

An easy fix for Cumulative Update issues is normally to simply to uninstall them, but in this case, the uninstall process will fail. It seems the only way to get rid of KB4559309 is to do a System Restore.

- so a restore point wouldn't have helped, apparently.

This is the last update that would have taken you to your build number:
KB4565351 CU Win 10 v1903 build 18362.1016 and v1909 build 18363.1016

To be honest, I've not seen a similar report (but there could be one); your Update History will tell you which updates to focus on (no screenshot - I did ask).

System restore is not an option for 2 reasons - a. vanished b. probably would have been ineffective if your problem was related to Edgium.

There might be an update you can uninstall, but that might not be possible for KB4559309

I found this:
How to uninstall Microsoft Edge Chromium - Your Windows Guide
but it's unofficial and probably needs intelligent adaptation.

Do you use disk imaging routinely as is so, so, so often recommended here - e,g, Macrium Reflect - free/paid + large enough external storage? If so, you can restore an image you previously created.

Are you sure you're logged in to your normal user profile? E.g. where does
%userprofile%
in file explorer's address bar take you? Check carefully.

However it also removed all my programs,

- how do you know? Have you looked at e.g.
C:\Program Files
?