How does the ABS pump effect braking under normal road conditions?

I have a '96 Honda Accord EX with a brake issue. I bought it two years ago with the ABS indicator light illuminated and it has never gone away as long as I've owned the vehicle and I've never really wanted to invest the money to get my ABS working.

Earlier this year I replaced a bearing on the front passenger-side and rotated the tires. Towards the end of the summer, I noticed that braking has been sluggish on wet roads, and that my front passenger-side tire begins to slip on gravel. Most recently, it seems my front passenger-side brakes have been grabbing too tightly, sending that tire into a skid even in good road conditions.

As I don't want to cause an accident, I have taken my car to a mechanic of good reputation. After my car sat at his shop for a few days I received a phone call from the mechanic telling me that his diagnosis of my issue is that the ABS pump has a leak and needs to be replaced. Apparently the circuit between the front driver-side and rear passenger-side isn't putting out any pressure, while the front passenger-side and rear driver-side tires are receiving pressure from the ABS pump. He is uninterested in pulling it apart to see if the leak is from the plugs, the accumulator, or any other part of the assembly.

I don't understand the brake system particularly well, but it is my understanding that when the ABS light is on, that means the ABS becomes wholly disengaged until the issue has been resolved -- even if it is just a faulty sensor. Is it possible that the ABS pump is creating the problem of my front passenger-side tire skidding under even the best of road conditions?

. Or is it just the caliper on my front passenger-side tire sticking.