The Space Shuttle was unique in terms of orbital servicing in that it had an airlock, a giant cargo bay, and CanadaArm.

The airlock allowed astronauts to go outside the vehicle to work on stuff without de pressuring and repressurizing the whole vehicle. No capsules have that capability and none are really large enough to add the capability. Remember it's not just the airlock but storage of the EMUs and any tools or equipment the task might need.

This leads into the cargo bay and manipulator arm. The Space Shuttle could get close to a satellite, grab it with the arm and then pull it into the cargo bay. The arm could also be used to maneuver astronauts out to a satellite to large to pull into the cargo bay. No capsules have these capabilities either.

A Dragon capsule has no ability to depressurize and repressurize safely and has no airlock. It's CBM compatible hatch is not large enough to fit either a NASA EMU or the Russian Orlan so there's no way for the crew to even perform EVAs. The launch suits worn on the Dragon and ACES suits worn on the Space Shuttle can handle emergency depressurization of the vehicle but are not EVA capable.

So it certainly won't be the Dragon, Starliner, or Soyuz doing any sort of manned in-orbit servicing of satellites.

I wonder if how hard it would be to build a space-only servicing vehicle that stays docked with the ISS.

The challenge is one of delta-v. To get from any orbit to any other orbit requires a change in velocity which requires propellant. Space is notably empty so you can't just go harvest yourself some propellant.

The ISS is in a particularly crappy orbit with respect to servicing satellites so a repair ship docked there would require more propellant than one in a better orbit.

The ISS is a nice destination for people because it's got consumables and room to move around. It's not a good pit stop on the way to somewhere else though. There's no spare propellant stored there so the only reasonable place to go after visiting the ISS is back to Earth.

You could launch another station with blackjack and hookers in a better orbit but there's still the propellant issue. Every time you'd want to service something you'd need to send more propellant from Earth. Just sending a dedicated vehicle up that could do servicing (like the X-37) would be far more efficient.

Or possibly a dedicated servissing platform, say with ion engines or refueling capacity. It could do slow orbit changes unmanned with crew launched only for the actual servissing, or possibly dragging broken sats for repair to a manned station and back - could be pretty workable in GEO belt if you manage to get and sustain a crew there.