I already own it on Xbox, but I wouldn't mind having a GOG copy of the game. I heard it was horrible when it was first ported, but they seemed to have fixed it? I don't want to have to download mods to enjoy the game either, so I just wanted to know if it's recommended to play this version, or if I should stick to the Xbox version. Reviews seem to be a mixed bag in terms of performance, but I see some are just a matter of different tastes.

Stick to the 360 version until the new PC patch releases.

I have played it quite a lot but the fact that the game was made for a console with such a miniscule amount of RAM and VRAM means severely annoying popping into and out of existance of NPCs. A clearly suitable case for an enhanced version.

Crashes during mission cutscenes happen quite often meaning eventually you'll be soft locked.

Is it Playable? Yes, it is!
To avoid Crashes use Powertools, and search on the this Site:
https://www.pcgamingwiki.com/wiki/Saints_Row_2

Is the gog version of Saints Row 2 playable? Yes.
Even the steam version is playable.
Have myself both of them. I got the dvd version of saints row 2, which turned out i had to activate it on steam.
Later on i also grabbed the gog version. Currently having the gog version installed and playing it sometimes.
However, by default SR2 is unstable. There is a fix for that. Not powertools or so, but a mod made by Mike Watson or better known as Idolninja. It's called Gentlemen of the Row. This supermod fixes car handling, making the game more stable and next to that has many more options. It's the swiss knife to SR2. You find it over at saints row mods.
https://www.saintsrowmods.com/forum/threads/gentlemen-of-the-row.24/

Read the instructions. After installing GoTR SR2 will be less wonky. I would still save on a regular base, as the game does have a habit of suddently crashing to desktop. Also, keep in mind that the dlc's were never ported and unfortunatly that will never happen. The announced remaster of of SR2 won't come. Mike would together with one called minimaul working on it, but mike passed away few years ago. Since volition is no more and nordic doesn't care then you can be sure it won't come.

I'm playing saints row 2 right now with no mods and its mostly stable.
I say mostly because whilst it will run fine for the most part, every hour or so you will have crash.
First was a hard hang that required me to move the process to another desktop so I can use task manager to close it, second time was brief gpu failure with a black screen that lasted for about 5 seconds before returning to normal.

Still better than the last time I tried playing the game on steam where everything moved super fast and then crashed.