Never owned a Surface, so can't comment on that, but I'm very happy with my One by Wacom (not to mix with Wacom One :p). It's fairly cheap as far as these types of tablets go, it's very responsive (I have 144Hz displays and it's so nice to use), has a nice sueface roughness, it's plug-and-play on Linux and has 0 maintenance (no batteries to swap).
What I like with my setup is that, contrary to traditional writing on paper, I can sit properly, looking forward, avoiding some bad neck and back pain I usually get otherwise.
(sorry, clicked Enter by accident and ended up posting this half-way 😅)
Yeah, it isn't a tablet in the usual sense of the word (i.e. it isn't a smart tablet), it's more like a tracking surface. The idea is that you use the little pen on it and the whole surface is mapped to your screen. There are differently sized devices, for different precision needs, much like A5 Vs A2 vs A3 etc. I have the medium one and I'm quite satisfied by it, but I had a professor that made class notes with the smaller model and it worked wonders too. Had mine not been offered to me, I'd would be more inclined to buying the small one.
They may be a bit weird to use at first, but I find that with you get the gist of it fairly quickly. I've had some colleagues try mine and while some got it faster and some had to spend a bit more time with it, they all got decent at it in a relatively short amount of time. I'm so used to it now that I make no conscious effort beyond what I'd do for traditional writing. I loose on a non-backlit surface and some of the physical pleasure of writing with true pen and paper (though the pen tip and tablet surface have a nice texture), but I gain incredibly productive superpowers in the form of undo, copy-paste, scaling and rotating, theming (love the white on near-black gray handwritten notes) and more (xournal++, for example, lets you embed images and even voice notes!). The pen even has nice pressure sensitivity, so you don't loose much expressiveness with your strokes.
Yeah, for this purpose, I'd say that device is not very well suited. The small version of One by Wacom is $40, which I consider fairly cheap for its quality and the value it can provide. In case that's too expensive, you may try the second hand market, I suppose.
Your Acer tablet may still be useful for other purposes, like a Plex/Jellyfin client or similar. For good note taking, even if the device functions decently well with Windows, I'm unsure if the touch sensors are good enough (even if they were originally, they may have degraded performance now, not sure) for a proper experience. Before I tried this pen tablet, I was quite skeptical of digital note taking, but now I love it, and it's mostly due to its incredible responsiveness.
I use Manjaro (based on ArchLinux) with KDE Plasma (now on version 6.1), though I use no touch interface, it's just a regular laptop onto which I connect this pen tablet via USB. For good touch support, you should look for the mobile variants of GNOME and KDE, namely Phosh and Plasma Mobile, as those are more optimized for that sort of devices. You should still be able to connect Wacom tablets and similar (there are drivers in the kernel itself).
Overall though, I agree with your last sentence, I think having the note taking tablet separated from the laptop may be better because you can just keep using your daily driver computer and, when needed, plug a fairly cheap but quality tablet and get a good handwriting experience and improved posture (very crucial to me)!
Happy to discuss this further!