Does their proposed patch panel costing about 4x what I was expecting actually seem likely to give any value? Is there are explanation for that cost?
I don’t know if there are “high tech” patch panels. They are mostly very simple, especially for use cases like yours. Maybe ask your builder how they justify the price?
For reference: Here in europe I could buy a panel with 24 CAT.6a ports for about 55€ on ebay. A panel for keystone modules would cost me 15€ plus 45€ for 24 CAT.6a modules. I would recommend to use keystone modules, because they are more flexible. The module is attached to the cable and then just clicked into the panel. This way you can easily detach or reorder the cables if needed.
Secondary question: is having a wall-mounted cabinet worthwhile?
Yes.
How will it work in terms of installing a switch and connecting from the patch panel to the switch?
You would mount both components in the cabinet and connect them with short patch cables like this: