Using outdated driver - What can possibly go wrong?

In your case, then you are running with no GPU acceleration at all whatsoever. Adobe has not supported that GPU at all for Premiere Pro since the CC 2015.3 (10.0) version. Therefore, your system is now permanently locked to the MPE software-only mode for rendering.

And your system's CPU is slow because it is also old. At the time of the FX 5800's introduction, even the top-of-the-line workstation CPUs did not have more than four cores and eight threads to begin with (this was when the very first Intel i7 CPUs using the Nehalem/Bloomfield architecture were introduced).

And the FX 5800 itself is now more than 12 years old at this point. Adobe does not want to support anything that's more than about four to five years old, especially when it comes to GPUs.

By the way, there had been two completely different FX 5800 GPUs from Nvidia that made it to market. The Quadro FX 5800 is a completely different GPU from the GeForce FX 5800 (the latter predates the advent of CUDA GPGPU processing by several years). The Quadro FX 5800 is based on the second-gen revision of the Tesla architecture (a.k.a. Tesla Refresh for the GT2xx chips) that debuted with the GeForce 8000 series GPUs in 2006 (the Quadro FX 5800 itself uses the GT200 chip that debuted with the GeForce GTX 260 and GeForce GTX 280 GPUs in mid-2008). In fact, the Quadro FX 5800 is a GTX 280 with 4 GB of lower-clocked GDDR3 VRAM (the GTX 280 had only 1 GB of GDDR3 VRAM that ran at a much higher clock speed).

So, get over it. The FX 5800 (and the GTX 280 that it is based on) were good GPUs in their day. But in today's world, they're no longer competitive (performance-wise). Adding further insult to injury is that the cheapo GeForce GT 1030 (a sub-$100 GPU based on the much newer Pascal architecture) can just about keep up with these TOTL Tesla-architecture GPUs overall.

I think you answered your own question:

performance degradation, instability and data loss"

I'm sorry to tell you this, but that FX 5800 is now completely obsolete. You see, Nvidia itself had ended all driver support for that CPU and all other Tesla-architecture GPUs way back in 2015! The last driver that was ever released for that FX 5800 was version 342.00, released on October 27, 2016. Accordingly, beginning with version 10 (CC 2015.3), Premiere Pro now no longer supports any Tesla-architecture GPUs at all for GPU acceleration.

And with each new successive major release of Premiere Pro, the program now requires much newer GPUs just to even run properly. In fact, the current 14.8 version of Premiere Pro now requires a GPU that has been released after 2014 (that is not based on a chip that was released prior to that year) in order to even be supported at all.