Connect Network Expansion Board/Patch Panel to Router
I haven't reviewed all your options yet, but the &quot;expansion board&quot; is a non-starter. It's a TELEPHONE expansion board and will NOT work as you expect. You can't just connect computers in parallel like phones.   What need there is a LAN SWITCH and all the appropriate re-wiring in your LV panel.   Maybe I'll add to this later...just wanted to let you know now that what you have won't work.
EDIT:  I just added a pic of my LV panel showing 2 small patch panels on the right and an 8 port switch on the left.  It's a little sloppy,  but it works.
Given the few number of connections in your home, you may not even need a switch if your router has enough ports on it.  The benefits of a patch panel is flexibility.