Every house I have ever owned had cold floors in the basement (even
when carpeted). The last house we bought does not have in-floor heat.
The basement is unfinished with the exception of being framed. We could
pour another 3" of concrete on the floor and put the right plumbing in
the floor for heat, but I am curious if there are simpler options

It sounds like you have a little extra headroom, so why not build an
insulated floor on top of the slab?

You could lay down a 6 mil plastic vapor barrier to keep ground moisture
from coming up into the floor. Then install 2x4 floor joists, install
insulation between them, and top them with plywood. You can then add
whatever floor covering you wish. If it were me, I would use pressure
treated lumber for the floor joists. No need to attach the joists to the
slab, just let the floor "float".

If headroom is a concern, you could switch to 2x2's as the floor joists, or
lay the 2x4's on their sides, and use rigid foam insulation between them.
In fact, you might be able to skip the joists entirely, and use adhesives
to stick the foam insulation to the slab, and then stick the plywood to the
foam.

Of course, this method won't heat the room, but it will provide a warmer
floor if you provide another heat source (electric wall heaters maybe).

This also assumes your basement is dry, with no water seepage.

If you are tiling the floor, another option is electric floor warming mats.
They get set right in the thinset when the tile is installed. Minimal
thickness, and it's a source of heat too.

Anthony