The correct place to install local user apps is

C:\Users\username\AppData\Local

Or even better to work with localization

%LOCALAPPDATA%

Steam is doing just fine writing into C:\Program Files (x86)\Steam\steamapps from an unpriviledged process. It just gives all users full rights over that folder. (whether that's a great idea is a different question, but UAC isn't a problem if you just set your access rights correctly)

Steam has definitely been handling Program Files wrong for as long as Program Files has existed. It's a marvel that there isn't more historic Malware intentionally targeting Steam install paths, though it looks like that storm is brewing on the horizon as I've seen references to such malware in the process of becoming a problem.

Steam should have worked to figure out a better approach to UAC years ago. At this point it is one of several big tech debt issues with Steam that has my confidence in Valve at an all time low as a consumer.

This assumes the user had admin privileges to begin with

If the user never had admin priviledges then you can't install in C:\ either and the program should just install in AppData\Local (as my sibling poster correctly stated)