I think mouse and keyboard only works properly in Desktop mode. Not sure if intended or a bug.
My suggestion is to open a support ticket with Valve/Steam. They might help you troubleshoot it, but most likely they will escalate the problem to their dev team as a feature request / bug report.
I second this if he can find a 2013~2017 for free/cheaper, I use an old Macbook Air with EndeavourOS mostly for writing, but my use case is not that far from OP's, it also runs some old games.
It is light and portable, and it is just as easy to find a refurbished option as the Thinkpad (at least in NA).
My only problems are with the drivers(the Wi-FI and the FacetimeHD camera) and the charging cable.
I tried a bunch of distros, but only with EndeavourOS I got them to work easily.
That said, if OP prefers better Linux support, and better cost x performance. He will be better suited with a Thinkpad T or X (T480/T490 or X280) refurbished will be in the price range.
X280 is barely above 1Kg, with a smaller screen.
T480 is chunkier, bigger screen and Ethernet port.
I think mouse and keyboard only works properly in Desktop mode. Not sure if intended or a bug.
My suggestion is to open a support ticket with Valve/Steam. They might help you troubleshoot it, but most likely they will escalate the problem to their dev team as a feature request / bug report.