How does it fare with modern LED bulbs? I imagine the placement of the reflectors assumes an almost-point source of light like a filament would provide, but most LED bulbs now have a plane of LEDs inside a frosted shell.

I guess would it make a noticeable difference to put an LED faux filament bulb in here? Or is it just not as fussy as I'm imagining?

I use a frosted LED bulb and don't see any reason it would work better with an incandescent bulb, i.e. it works great.