From among many different options mulled over by NASA, the decision was ultimately made to design the International Space Station to provide the most pristine zero-gravity environment possible, in order to facilitate experiments involving everything from physics to the life sciences. This single requirement alone creates enormous architectural challenges, which have compounded the cost and complexity of the project.
The main issue with satellite repair is that your service platform must start in the same orbit plane in which the satellite orbits. The station is a potential service site only for a satellite that is, for instance, in an orbit 51.6 degrees inclined to the equator, among other exacting requirements. The chances are slim that any particular satellite meets these conditions. If you think of a satellite in orbit as a runner on a running track, it is clear that you can only rendezvous with and stay with the runner if you, too, are on the same track. Changing altitude and phasing with the satellite is simple, like changing lanes. However, every "running track" in space is at a different orientation. Jumping from one track to another takes a lot of energy. Even a one-degree orbit plane change would cost the station nearly 12 metric tons of propellant, which is a greater mass we'd have to launch in a refueling mission than the mass of almost any spacecraft we'd want to fix.
Most satellites are much farther inclined to the station's orbit than one degree. Similarly, most satellites don't have the propellant available to change their own orbit plane, so they can't come to the station. A tug would require a lot of propellant for two round-trips—one trip to fetch and a second to reposition the satellite. Since most satellites fly much higher than the station and are in vastly different orbits, it would take too much propellant mass to consider changing orbits to retrieve them. (Note: the very limited time that a launch site lines up underneath the orbit plane of a satellite or a space station is the reason why there is such a short daily launch window for a rendezvous mission. We simply can't afford the propellant to change orbit planes if we don't get it lined up right in the first place.)
Working on a real spacecraft in a shirt-sleeve environment inside a space station presents real problems. Every spacecraft deploys valves and other instruments using pyrotechnic mechanisms once they reach orbit, to prepare for further flight operations. It's not wise to have unexploded pyrotechnics inside a shirt-sleeve environment, but it is similarly dangerous to have delicate antennae deployed, or worse, opened hypergolic propellant isolation valves—which lock in fuel that ignites when two components are mixed—with the spacecraft inside. The air inside the station is carefully conserved, so any toxic risk (like propellant) is potentially catastrophic. Astronauts do not have the same unlimited supply of fresh air that workers on the ground enjoy.
Even if one chose to accept the risk of handling a spacecraft inside a human-occupied sealed vessel, we would face the final problem of how to build a chamber wide enough or a doorway strong enough to get a large spacecraft inside. The aerodynamics of launch vehicles limits the diameter of the payload to about five meters. Even assuming you chose to work with only a few centimeters' clearance between the walls, we would have the problem of getting the spacecraft in and out the end of the chamber. A five-meter hatch against atmospheric pressure must withstand nearly 200 tons of force, meaning that the bolts and seals would have to hold one ton of tension for every 16 centimeters. This leads to some massive framing and bolts (or hooks) that would cause even a very crowded hangar to be a difficult thing to design, build and launch.
Although the prospects are much more favorable for using the station as a staging base for a further deployment of spacecraft that leave Earth's orbital environment—like the Webb telescope and missions to the moon or Mars—such outfitting would need to be done in the vacuum of space with the spacecraft docked to the station. The launch window to the moon, Mars or other points would occur far less often on the station than from the ground. For example, a lunar launch window from the station occurs approximately every 10 days. Mars is even more rare. From a rotating Earth, we can choose a launch window every day that aligns us into the plane of our trajectory to the moon or desired planet. The moon's 29-day orbit around the Earth helps to bring the ISS into alignment with it every 10 days, but Mars' relative motion is so small that we must wait for orbit precession alone, which gets us aligned roughly every 30 days or so.