That's always a hit and miss thing, and usually depends whether the latest drivers written for the game are as well written specifically for new GPUs and OS updates. They try to encompass all needs, but I'm not going to lie to you, GPUs several gens old are not exactly high on the list of priorities when it comes to testing drivers.
So I can see why the latest drivers don't play well with your R7 360, but you have to realize, the older drivers also aren't going to have much for today's games or OS/API updates. This is particularly the case with AMD, because most games now are Nvidia endorsed, so AMD are at the mercy of the time the devs give them for consultation.
Curious as to why you even have such an old GPU in the system, and what good it's really doing if you have to use such old drivers because of it? If I were you I would make a couple sets of benches, one running with current build on the old drivers, one with the the newest drivers with the 360 disabled, and compare notes.
I've spent easily as much if not more time on ATI/AMD GPUs as I have Nvidia, and my experience is, the longer you hold on to your GPU, the better the result with Nvidia driver wise. I currently have a GTX 1080, which is a 3 yr old gen, but I can run the newest drivers no problem on any game.
On a side note, there is a slight chance this may or may not be W10 build 1903 related. I've read of some people having audio and video problems on 1903, which is a big part of the reason I've not updated to it yet. Some claim if anything it's faster than 1809, but it may also depend on the hardware and software you're running as far as performance and compatibility goes. The fact that you are saying it's a constant crashing vs just performance thing kinda indicates it could very well be more than just the driver. Have you gotten on the AMD forums to see if others are having such problems, especially those on old GPUs with the latest 1903 W10 build?
That said, AMD drivers are generally more finicky to install, especially if they include any big updates for ReLive, or the RT support they've been hinting at (not sure that's even a thing yet for them). It's possible you need to reinstall, and it's usually recommended to do a clean install, which involves a complete uninstall of the old driver, then install the new ones. There are several 3rd party tools for this, since AMD's uninstaller is never very thorough, but make sure you use a reliable one. Less thorough is always safer than too thorough. Many use DDU, but I found it left driver bits behind. I actually had better luck with Driver Sweeper. It's no longer supported, but you can still get the 3.2.0 version, and it removes everything without removing too much.
https://www.techspot.com/downloads/4266-driver-sweeper.html