If the EGR valve is stuck open, or the cat is severely plugged, you’d definitely notice it. You’d get poor acceleration, missing, rough idling, and stalling. Since you didn’t mention any of those symptoms, it seems unlikley the EGR or cat is related to the xmission problem. Still, it may well make sense to fix whatever is causing that problem before trying to fix the xmission.
There’s a couple of simple things you might ask your mechanic to try first. Make sure the fluid is reasonably clean and at the fill-mark. Consult your owner’s manual if you’ve never checked the auto xmission fluid level, as you have to follow a specific sequence to get an accurate measurement. And look for any signs of deteriorating rubber vacuum tubes that might be leaking, esp for breaks/cracks at the connections.
If somebody recommends an xmission flush, from what I hear said here, that’s probably not the best thing to do. It’s better to have a complete xmission service by-the-book done instead.