With a somewhat decent length ethernet cable, it is actually not too messy.
Depending on the power draw, you may want to make sure your switch also supports PoE+.
I have tested both PoE HATs and PoE USB Dongles on both RPi 3's and RPi 4B+'s and haven't had any issues. In that testing, however, I would caution against some of the PoE HATs that are made for RPi 4B+'s because some of them barely put out enough power and the one that did got quite hot.
In the end, in both the RPi 3 scenarios and the RPi 4 scenarios I ended up using the USB PoE splitter in conjunction with a PoE or PoE+ switch, run over a CAT 5e or CAT6.
In terms of quantity with which you can get away with... I can't speak to an upper limit really, as I was just fine running over 40 RPi's which used a handful of Unmanaged Switches.
In a couple instances during development and testing, I used Managed Switches just for the sake of organization. It starts to get a little tricky trying to track which port goes to which RPi...