Drywall – Can drywall harden or become brittle over time? Or, am I dealing with plaster

Freestanding steel studs are fine (assuming they are anchored to a sill and top plate). Steel studs anchored to the furring strips are fine. Why wouldn't steel studs anchored to the furring strips, plaster and lath be fine? The only limitation would be if the lath and plaster had significant bowing or hollowing in spots that the new studs came in contact with (minor hollowing would not make much difference).

Also, as you install, you want to make sure that crumbling plaster does not get wedged in a way that prevents the studs from creating a smooth planar surface.

While plaster is not much of an insulator, you still have a bit of air trapped between the plaster and the brick in the furring strip troughs. Add high density batts or rigid foam between the steel studs to improve the R value.

If you are very concerned about insulation, wood studs are poorer transmitters of heat and should be considered. Also, creating a real thermal break by not having the inner wall directly abutting the outer wall would improve the situation. But this approach does eat up interior space.

The best way to handle this situation is to span two studs at the end of the closet with a smallish plank, screwing it into both studs. Then you can screw the rod end mount to that plank and rest assured that it's as sturdy as you can get it.

The "far" end of that plank will very probably screw to the stud at the far rear corner of the closet's end wall, which is kind of nice because that hides almost the entire plank.

An alternative, but one that's only suitable if you're definitely never going to hang a lot of clothing from that rod (clothing weight accumulates shockingly quickly), would be to use drywall anchors to attach the rod end mount to the drywall.