...or if they can, why so rarely?
I mean: instead of placing a satellite of nondescript shape inside a fairing which is discarded somewhere above the atmosphere, build it in such a way that its chassis doubles as the fairing, only small parts ejected or opened to e.g. extend solar panels, radiators, antennas etc. Simply, the nose of the rocket enters the orbit and serves as the satellite.
You gain a lot of volume and quite a bit of mass of the payload. It can be shielded better against radiation and small space junk. Its cost will be lower than of classic payload + non-recoverable discarded fairing. There would be some loss of delta-V due to extra mass, but the massive extra volume could hold fuel that should well make up for it; never mind it could be inseparable from the transfer stage and just carry it as unimportant extra mass to the end-of-life, saving up on own propulsion system.
Does the fairing add so much mass that it's essential to discard it during the last stage of acceleration before MECO? Or are there other reasons why satellites aren't just made to fly as the top of the rocket, but packed inside it instead?