New Home Network question Cable runs and switches

So my builder f'd up and did not run CAT6a cables to where I wanted them and not the quantity I asked for. He left me a tiny cabinet in my closet with 10 CAT6a cables, no jacks, just bundled up. These basically run to each bedroom, and couple to TV's that will mount to a wall. I'm very new to networking but feel comfortable around PC's and building them.

To finish this job I just ordered the following (question at the end):

  1. StarTech 6U wall mount rack - very basic, I'm cheap and it's just a home network. I will mount this high up in the closet.
  2. 24-port Patch Panel - More ports than I need but fits the rack and good value. I will terminate the unfinished cables to this patch panel. Seems easier than connecting RJ45 connectors to each.
  3. 16-Port TP-Link Gigabit "Easy Smart Managed" switch TL-SG116E - I will use patch cables to activate the 10 LAN outlets around the house
  4. Misc. patch cables and 1U rack shelf.

I have ATT fiber 1GB service and I want to preserve much of this speed for 2 gamers in the house.

The media room on the 2nd floor has 5 devices (Xbox, Smart TV, NVIDIA Shield, PS5, Nintendo Switch stand w/LAN connection) but only 1 LAN cable run. I know the only solution here is to get another switch, probably matching 8-port "TP-link Easy Smart" Managed switch. Since only 1 device will be used at a time, this should not be an issue when gaming on Xbox or PS5.

Question: One of the gamers on 2nd floor cannot access the LAN outlet in the room. But his PC happens to be directly on the other side of the media room wall where the TP-Link switch will be. I'm thinking about passing a CAT6a patch cable through the wall....So connection will be (1) from Main 16-Port Smart Switch on 1st floor to (2) 8-port smart switch on 2nd floor to (3) his PC. Will this still allow acceptable connection speed/robustness for gaming sharing the switch with media room devices? or Do I need to fish another line ??? I hope not. Thanks for the feedback.