Plumbing – Can a shower mixing valve leak water from cold to hot lines if there is no pressure on the hot line

I assume when you drained the tank, you turned off the main cold water supply. I suspect that the cold water valve is not opened all the way now. Check to see if in fact it is open full. If it is, and the pressure is still low, you may have a jammed or damaged valve. This is very common on old valves that are not used very often. The seats and washers inside the valve can degrade over time and now are not allowing water to pass freely.

From your comment "re-plumbed for code reasons" and the odd behavior you're experiencing, I suspect the hot water circulation system might have been patched/crossed into the cold water system and/or the pump is faulty.

For a hot water circulation to work, the hot water pipes must be in a closed system (other than the obvious exit points like a faucet, dish or clothes water connection). The pump will draw hot water from the tank and circulated it back into the tank, for as long as the pump is running. You'll always have hot water on demand.

The pump is usually a passive device, meaning if the pump is off, water can freely move in and out of the pump without obstruction. But if you're not getting hot water when the pump is off, the pump maybe faulty.

And you want to check to see if any hot water pipes got crossed/connected into cold water pipes. I suspect they are.