Generally-speaking, USB mice/trackballs don’t need drivers.

Except to program the buttons

No, like...okay. How do I best put this?

"Driver" software is going to be software designed to talk directly to hardware.

In Linux, if you go looking for "driver software microsoft trackball", you probably won't generally find what you're looking for to bind things to mouse buttons.

In Windows, it's common for you to buy a device and for it to be bundled with some software written by the device vendor. That's because the device vendor is writing the software to sell their product. They're selling you a package of software and the device, and they're bundling the software specifically because they want the hardware to sell. While technically the "driver" part is only the bit that talks directly to the hardware, and often there's a lot of other software bundled, it's not uncommon to use "driver" more-colloquially to just refer to all the bundled software.

In Linux, while sometimes vendors do release proprietary software with their hardware, you're more-likely to be using software from some open-source projects. Those projects don't care about how some specific piece of hardware does in the market. They just want the functionality to be available everywhere. So unless a piece of hardware is truly unique or requires some way of specially interfacing with it, you're more-likely to be using a software package that works with many different pieces of hardware. They won't call it a "driver".

In general, for things like mice, the hardware-level stuff is all in the kernel already, because the USB standard already defines how those devices need to act to present their extra buttons; it's not necessary to add additional special software to talk to the hardware. Instead, you're just looking for a piece of software that can send a series of keypresses or whatever when you click a button, something like antimicrox.

Windows games that have integration

not games, other than the profile layout, but system monitors in windows display on it.

Fair enough. Just wanted to moderate expectations. I vaguely recall that Logitech had some other full-size "gamer keyboard" in the past that had a text display with some sort of game integration. I think I remember Teamspeak integration being on the list.

kagis

I don't think that this was it; what I'm remembering was an older keyboard, but:

https://www.amazon.com/Logitech-Programmable-Gaming-Keyboard-Display/dp/B001NXDBI6

A color GamePanel LCD displays game stats, system information, VOIP communication data, video playback, image slideshows

I'm guessing that the VoIP there is talking about Teamspeak, and they mention "game stats".

And I don't think that there's an out-of-box way to have something like that running under Linux for your G13 and displaying specific-game-relevant information.