You could use a structured media enclosure like Leviton sells but it's not deep enough if you want to install a bunch of active gear in it. It's fine for a patch panel, a switch, a cable amp, a modem, and even a small UPS if you get the extension bracket. Mounts in the wall. But if you want to start installing NAS units, it's not a good choice. But it's a neat, unobtrusive solution.
https://www.leviton.com/en/products/residential/networking/inside-the-structured-media-enclosure
Yes. I had something very much like this in my previous home. It did the trick at the start, but when I started adding to it, it didn't grow very gracefully. Each addition was overpriced and not exactly what I wanted. I could go this way, but it would not be my preference.
But then to achieve a neat solution, you're looking at $65 just for the punch down for 12 Cat 5e ports with the 476TM-512 , and $145!! for a 24 port Cat 6 version. That's heinous for what amounts to a branded patch panel.
I've got a similar issue as the OP, except the builder left all the wires hanging inside a structured wiring box. Helpfully, there's no slack at all for the Cat5e, so to avoid dropping $145 as above, I'm going to have to mount a patch panel vertically right next to the wiring enclosure just so I can then use patch cables to reach a switch on a cheap-o open rackmount setup with a 24 gigabit switch that will only run $100.
Your sentiments are pretty much exactly what had me going in the direction of something a little more open and flexible (even generic), versus proprietary stuff. Plus, I already have the switch that would never fit in that.