(I haven't gotten through everything yet, so full reply pending, but here's a status update).
In browser mode it doesn't run at all. I think it's because I don't have Unity Web player installed (and I'm not sure the beast exists for Linux at all - all I find is ooold requests for one).
I've (reluctantly) rebooted to finally give the NVIDIA-blocking version of /etc/modprobe.d/blacklist.conf (pinched off a friend with similar setup) a spin:
blacklist nouveau
blacklist drm
blacklist drm_kms_helper
blacklist ttm
blacklist mxm_wmi
alias nouveau off
(some of the modules don't seem relevant, but I'm told they very much are. I took his word for it - though he's forgotten the reasons, I never knew them :D )
Intel-only rulez! Now mouse feels more like "a computer with slightly different speed settings for pointer movement than what I'm used to" and it doesn't take ages for Clam to cross a room or for me to aim at a menu line. So, definitely looks like "CPU can't do that much software rendering emulation, need hardware GPU - even 'slow' Intel will do".
Still don't know what's so computationally-intensive and am sure there's a possibility for optimisation/low-quality mode, but the work-around worked.
Which, of course, means a lot more of things to talk about, but not quite now. Just checking in and letting other unfortunate souls know about a possible solution :)