It's a lovely room.

We've installed recessed lights in our basement with flat ceilings. I do know they make some quite small housings but I don't know how you would set them in the curve. I agree most solutions you see online are for slanted flat ceilings.

Another option might be to install one center fixture, you could maybe use a ceiling medallion and fill plaster in around it so it looks more purposeful?

I do have an old home that has very minimal overhead light - our living room actually has no lights overhead. We use floor lamps and wall sconces. With LED bulbs of the right temp, and the right fixtures it's plenty bright for evening relaxing and reading.

no, i would not . I would like to know how you use the room.

Also for older homes, does it make sense to install recessed lights? Thanks!

Sure if you want to remuddle the home and lower its value for a future sale.

Please, do not install recessed lighting in that beautiful curved ceiling. I have an older home, and the living room has no ceiling fixture. I use floor and table lamps. If you feel a ceiling fixture is absolutely necessary (and you are determined to spend money you don’t need to spend) install a center ceiling fixture in a (vintage) style that is appropriate to the architecture, and please- NOT something contemporary from a big box store. Perhaps looking at photos of rooms of this style and vintage will give you some good ideas.
You have a beautiful older home with great bones- embrace it and enhance it; don’t dumb it down with a remuddle, please.

The ceiling is a unique feature. The best way to light that is with uplighting along a reveal at the perimeter (hidden by the crown). You could put in a couple recessed lights accenting the fireplace, but I would use a very expensive adjustable housing with a flush drywall trim. The cabinet line would also be nice with some accent lighting, but those cabinets appear to be DIYed and not really matching the rest of house, so a long-term plan would be better for those. So on the last 2 counts probably no lighting at this time, but you could investigate whether a tape light behind the crown would work.

Moving down from the ceiling plane this room style would work well with a couple wall sconces if you don't want all floor/table lamps.