I went with AMD for compatibility playing games, but AFAICT AMD ROCm is not in a great state for computation. Why can't I have both?

That's something like what they're describing as "a nightmare," isn't it? "As agreements and law allows," is part of the nightmare. Under a modern OS, it should not be difficult to have the full capability of the hundreds or thousands of dollars worth of hardware you paid for.

NVIDIA drivers work perfectly fine, when you decide from the beginning to use them, instead of attempting to first use alternatives like nouveau.

The only problems appear when your Linux distribution decides for some reason to make difficult for its users to choose the NVIDIA drivers, like GUIX. On distributions where the users can choose NVIDIA freely, like Gentoo, which I am using, there are no problems whatsoever with NVIDIA. I have used the NVIDIA drivers for Linux and for FreeBSD during more than 20 years, on many kinds of desktops and laptops, with no problems. All this time I have also used Intel GPUs and AMD GPUs, and especially with the latter there have been problems more frequently.

Unfortunately, some of the Linux kernel developers actively try to sabotage the NVIDIA kernel driver from time to time, e.g. by imposing restrictions on the kernel symbols that it may import, which is certain to complicate the work of the maintainers of the NVIDIA drivers, and which can cause problems for the Linux users who are not aware that for this reason they must have a version of the NVIDIA driver that is matched to their kernel version.

I much prefer to have only open-source privileged code, but it cannot be said that NVIDIA has not done a good job with their Linux (and FreeBSD) support, which has been much better than that of almost any other hardware vendor. Only Intel had even better support for their hardware, including not only CPUs and GPUs, but also networking, WiFi etc. However, even with Intel, their Linux GPU drivers have been frequently worse than their Windows GPU drivers (like also with AMD), unlike NVIDIA, where their Linux drivers always had the same performance as the Windows drivers.

While non-professional users may not encounter many problems with the Intel or AMD GPUs, any user that has needed more complex OpenGL applications has frequently encountered problems in the past when the Intel or AMD OpenGL implementation in their Linux drivers was incomplete or buggy in comparison with NVIDIA. Such problems were easily exposed by just running popular benchmarks like SPECviewperf or the Unigine benchmarks on Linux and comparing what you see with an NVIDA GPU with what you see with an AMD or Intel GPU. The Intel and AMD drivers have improved in recent years (or more accurately, Mesa is what has improved), but previously it was hard to use under Linux non-NVIDIA GPUs for any really demanding graphic application.

Modern NVIDIA drivers for their more recent cards are actually okay, again.

AMD cards need propietary firmware.

Honestly back when I was still in college one and a half decade ago, it was quite clear the whole Nvidia-only ML and AI libraries weren't a good idea