but I feel like it’s not working with the integrated graphics card

That’s possible… but it’s also not exactly clear what “feeling” you have about this, and I don’t know what other graphics card it could be using? I don’t really understand this, are you just saying the performance is bad? That I would believe as a possibility due to the distros you’re using, it’s probably fixable with the right twiddling of knobs but whether you want to do all that tinkering is a question I’ll address later. First, let’s address the elephant in the room:

You typically don’t need drivers from a website for Linux, especially not graphics drivers and if you do the OS should be able to get them itself. Which drivers to use are notoriously finicky because they tie in so tightly to the OS itself, and there are competing proprietary drivers that might interface with the hardware better and suck at interfacing with the OS and kernel, and open-source drivers that interface with the OS and kernel perfectly but sometimes suck at interfacing with the actual hardware, and the tradeoff of which is better for a particular OS or particular kernel or particular hardware is really not always obvious or intuitive and changes over time.

In short, I personally find this is a good area to trust the distribution you’re using is picking a good option for you and will provide reasonable alternatives within its own packaging system. Assuming you’ve picked a good distribution, you don’t need to mess around with installing proprietary drivers from the websites manually which tends to just make a mess of your whole OS, which brings us to the next topic we need to address:

Mint and Kubuntu are nice comfortable stable “desktop” variants but they’re not really optimized for gaming, and gaming on Linux is a space that is in very very active development right now and one where it really pays to be on the cutting edge, because projects are improving things rapidly and you’ll only get the benefits of those improvements on the bleeding edge gaming distributions that are quickly integrating those changes. Otherwise, you’ll be stuck on a “stable” distribution that might be years behind graphically, and years is a huge amount of time in Linux gaming at the moment.

While you might think “stability” is an obviously good and important thing to have, the reality is it also means you’re not getting improvements, and sometimes those improvements are really good or even completely necessary for modern and esoteric hardware support, like the kind of modern and slightly esoteric hardware you have. It’s also a bit of a misnomer, all distros try to be pretty stable as far as not crashing or corrupting. It’s not something that commonly happens even on “unstable” distros. Unless you’re using something that has very hard coded environment requirements and dependencies, you’re not likely benefiting from the kind of “stable” that stable distros provide anyway.

A lot of people recommend starting out with Bazzite as a relative newbie to Linux who’s interested in gaming. It’s a pretty safe distro and gets around the stability of crashing vs the stability of the software environment by essentially giving you “snapshots” of each new version that you can choose between or go back to the old version if it’s causing trouble, similar to Windows system restore, but better. It should have good performance and get you quickly and easily set up for all the gaming and media you can handle.