My laptop (running Mint), is a hand-me-down that only supports OpenGL 2.2 with it’s integrated graphics, and so I installed Mesa, but I do not know how to set things to use Mesa for OpenGL, and so it doesn’t really help right now.
I see you mentioned the cause, nevertheless you are still running, at least partially, into a known problem:
Are you sure, that using Mesa is the solution to your problem? Perhaps there’d be a better solution in your specific situation.
However, if Mesa is really the solution, you can look here:
Yep, 100% sure my laptop only has intel graphics, it was my parent’s old work laptop from 2009 it was clearly low end, I was surprised Half Life ran at good speeds.