Yes, more details/pics please.
My McMurry is like the one shown at the link below. Its the only model I'm aware of with that name spelling. It was made by Doc Freeman's, a chandlery in Seattle back in the 70's and 80's, and ended up on a bunch of PNW/west coast boats.
If that isn't what you have, ignore the rest of this post.
I think what you are calling the "chainplate" is the wildcat. It is the "chain gear" that grips the chain and pulls on it. (A chain plate is a flat metal piece that is used to hold mast rigging in place on a sailboat.)
On my McMurray, the gypsy can be loosened to freewheel or tightened to stay fixed to the shaft (and only move with the motor) via the handbrake. If the handbrake is tight only the motor moves the chain, if it is loose it can freewheel. Because the handbrake is so small and hard to tighten, I don't as a practice use it to drop anchor/chain, its too hard to control and too hard to get tight again. I "manually" put my chain out with the windlass motor. My anchoring is usually pretty controlled and methodical so this works anyway, freewheeling would actually almost be self defeating as I don't want a pile of chain out, the windlass motor goes almost too fast anyways. We typically set out enough to hit bottom + 15 ft or so, then as I start to back down Y-von starts putting out the rest of what we've already agreed on (we have our all chain rode marked every 25 feet so we can set a pretty close amount to what we need.) My windlass is wired and configured with two foot switches, one to reverse and one forward for anchor up and anchor down, chain in and out.
You note you have greased the "rode wheels". The only lubrication necessary on the McMurray shown in the link above is the very particular oil level inside the housing. Are you talking about the friction plates shown in the drawing?
The friction plates on either side of the gypsy should be clean and dry so they will grab the gypsy when the handbrake is tight.
A question for you - does your windlass (has it ever) work to power down your anchor? These windlass' are powered by a motor that is basically a modified Ford tractor starter motor. Some people over the years have replaced the original motor with that starter motor which only spins one direction, losing the ability to power down the winch. If that is the case there is at least one company that produces a version of the motor that reverses. I have not purchased one but have the research info (and have recently confirmed it) on where to get that motor.
Additionally, there can be problems with exactly how that motor is installed back into the winch housing and the overfilling of the oil, causing oil from the housing to get at the motor innards which could be starting to cause your intermittent motor issues. (Of course, those could be 1001 other things, foot switch, wiring connections, relays, battery, etc etc....)
If any of this sounds like it could be your windlass let me know and I can give you everything I have gathered...