Disclaimer, I'm a learn as I go. With the help of the smarter guys here I'm becoming more hands on. More seasoned guys please jump.
1st I'd check the pin hole in the operating valve to see if it's plugged. Wouldn't that be fantastic if that was the root of the problem.
2nd I'd continue to flush the system until no debris shows up and the oil is clear and clean. (Even if it doesn't fix the problem, what's the down side to having debris free clean fluid.)
Food for thought. I believe the 3 types of oil we use are 20w-50 motor oil (which is a detergent), 30w non detergent and Redline MT?
I used 30w over and over for the flush because I thought it was the correct oil per recommendation of Healey Surgeons web site and phone call to Bruce. From their website:
Fill the transmission with 30 weight NON DETERGENT oil. If this isn’t available at your local auto parts store, machine shops will carry it. Or we can supply it. Change oil after an initial test drive and has gotten good and hot.. It will need 3.75 quarts for sideshift and 2.75 quarts for centershift
BCS has an orange Austin-Healey Tech Talk Technical Article Book (I use all the time and recommend) that states on page 23, 20w-50w is satisfactory or to be correct, 30w non-detergent. (photo attached of that from the book). HOWEVER, I believe David Nock of BCS now recommends Redline.
My point for your situation is flushing the system with 20w-50 seems logical because of the detergent. Healey's have been using it for a very long time with no issues.
I also wonder if you can remove the plug at the base of the pump and by turning the driveshaft, see if the plunger moves.
Sorry, no. You'll see the bottom of the oil pump body (moss 97)
Do you think it is very likely that the pump or the ball valve just 'went bad' all of a sudden?
Seasoned guys, please field this one.
Does the cam follower and return spring come out through the bottom with the pump?
Cam follower aka oil pump cam (moss 92) is slid on the main shaft, and held in place when the the OD and Transmission are pressed together.
Return spring (moss 96) is part of the oil pump plunger and will come out with the unit from the bottom of the OD.
I'm worried it won't and then I'll have to take the OD apart.
If it come so that, feel free to PM me. I've had the tranny/od out many times and there are some some things that make it easier once you know how. You don't need to take the OD apart, just remove it from the transmission. Removing the adaptor plate (moss 11) from the transmission is the best easiest way. Then you'll be able to see if there is any damage to the pump or cam and confirm pump moves. Putting OD back on, well it's heavy and you need the mainshaft spline to line up with the OD spline. Then use the string procedure to keep the oil pump plunger roller down so the mainshaft will not contact it allowing the shaft to enter the OD.
Hopefully flushing the fluid will dislodge any debris, gunk or stuck parts.
You may look into the accumulator housing assembly.
Best of luck.