Advice on setting up home ethernet (with unused cable already in the walls)

I live in a townhouse (built in 2002, if it matters for context)

it has ethernet cables in the walls, that have apparently never been hooked up (yay futureproofing)

they’re cat 5e cables (I checked on the cable sleeve, because I wanted to make sure it would support gigabit ethernet and not just 100mbit)

behind each wall plate, the ethernet cable is just coiled there, not terminated in a connector and not connected to anything (along with coax cable and telephone wires which are hooked up to the wall plate, but which I’m not currently using at all)

the cables run to a wiring box in one of the bedroom closets. here they are also just hanging around, unterminated and unconnected to anything.

I have a rough idea of how to DIY this, but I've never done it before - the extent of my networking knowledge is layer 2 and above. so I'm looking for any protips of the sort that you figure out after doing it several times but that aren't obvious the first time you do it.

right now, my shopping list is:

• a patch panel (I’m eyeing this one) which will terminate the cables in the wiring box
• a punch down tool (maybe like this one) for…umm…punching the wires, it seems like?
• a gigabit switch (I have a spare 8 port one that I’ll use, there's only 6 runs of cable total) to go in the wiring box
• a wall plate (like this) for each of the 6 endpoints

I really only care about 2 of the 6 - the motivation behind this project is that my modem & router are downstairs, my home office is upstairs, and currently I run powerline ethernet between them. powerline ethernet isn't great, but it sucks especially hard when something like a portable AC unit is running on the same circuit, and that's currently making the internet speeds in my home office suffer. but the 6 cable ends in the wiring box are unlabeled, and so it seems easier to just wire them all up rather than play guessing games to figure out which of them are the 2 I care about.

the main thing I'm unsure about is the termination of the cables with the punch-down tool. I've crimped ethernet cables, years ago, and hated it, due to having clumsy hands and large, ungainly fingers. this doesn't seem quite as bad, but I'm still cautious about having to do all this in the fairly cramped closet space, and with limited ability for "do-overs" due to the finite amount of cable installed in the walls. I'm thinking I may buy a cheap ethernet cable and sacrifice it for some test runs of the punch-down tool.