Okay, this can be fixed really easily. I assume that the explanation for the problem is here: https://bugs.launchpad.net/hplip/+bug/1576800/comments/13

What worked for me was, simply, removing HPLIP and the related package, hplip-drivers, in the Software Center. Next, installing it again. Now both the HP Device Manager and Simple Scan work fine.

I can say that I have been having similar issues in both Windows and Linux (Solus, Ubuntu, and Arch) with the DeskJet 4765 since around the time of this update, however I did not have time to do anything about this (on Windows I had simply replaced the drivers with an older version that I had previously backed up), since I personally use the printer less than my family, and Windows 10 no longer needs driver installation to get them working, both wired and wireless (this is a WiFi Direct all-in-one).

Recently I had picked up an OfficeJet 6500 (no wireless here) which someone had thrown out, however it scans without issues (tested on Windows 10), I don't have cartridges for it, so I cannot test printing, however it appears as it should under Linux.

Both exhibit the same exact issues under Linux, they show up as scanners (as well as the rest of the functions) in gnome-control-center, however they fail to appear in any and all scanning software from the package repository.
Printing from the DeskJet 4765 work as it should, both wired and wireless.

VueScan works flawlessly, as it utilizes it's own driver database, however it is both closed-source and shareware (their watermark is obtrusive in my opinion), so it's not a choice for me.

Removing and reinstalling hplip-drivers (I do not use HPLIP's GUI) yielded me no results, my sister's laptop there was no need for installation, since there it was a clean install.

Unfortunately, to get scanning to work properly, you need to install hplip, and set up the printer using the browser-based CUPS interface (or use the CLI), and even then it may take a few a few tries until scanning works properly.