I have worked as a car mechanic, restorer, welder and auto-electrician.
You can drive your car after it fails an MOT if your current one is still valid. End of.
If your MOT has run out, you can drive it somewhere for pre-arranged repairs, or for a pre-arranged retest, but only if the vehicle is not dangerous to drive and not 200 miles away. This is open to interpretation by yourself and the Police. Just be reasonable about it and use your common sense.
The MOT tester can advise you not to drive if they believe it to be dangerous to do so. They cannot prevent you from removing your property.
A failed ball joint probably means it has some play or split rubber.
My car failed its last MOT on a cracked
spring. It was only cracked at very end so it was fine for me to drive it away and repair it myself. Had it broken in the middle, it probably would have been a good idea to have it towed. But then I'd have noticed well before MOT anyway!
MOT garages where they do repairs do tend to talk crap and downright lie to make a quick buck. This is why I ALWAYS take to testers that only do MOT's. They have no incentive to lie.
I have worked with people who will find any excuse to make money out of you by lying, cheating and stealing as far as your car is concerned. Better educate yourself and avoid these tricks.
I know of a main dealer customer who got his Nissan serviced and tested every year by them from new. His car failed an MOT on a handbrake. They quoted him £720 for new rear calipers. After discussing it in detail, I advised him how to adjust his handbrake, then told him to go to a VOSA MOT centre. It passed first time. And this they did to a loyal customer. Think how much they must have been fleecing him over the years?
Having said all this, it is still your responsibility to make sure your vehicle is roadworthy, regardless of an MOT. If the ball joint was broken completely, you'd know about it for sure anyway.
If the vehicle had dangerously defective tyres, suspension, brakes or steering (or any other structural damage likely to affect any of the above), then you can be liable for a fine and 3 points for each defect.
But seriously, you don't need to worry about play in a ball joint stopping you from driving away a reasonable distance and getting it repaired within a week for a retest. This is also why you have 30 days to get your car tested before the due date and have your MOT run from the date the current one expires. It gives you time to get parts and repairs done.