Unless the motor mount was *really* loose, I'm not sure it would cause the car to pull one way or the other. If the rubber engine mount has *completely* failed, you may be able to push on the left or right side of the valve cover and move the engine significantly.
Right/left pulling is usually a suspension problem....typically on a Midget this would be on the front suspension (loose wheel bearing, worn king pin, lower A-arm bushing/pins worn out...possibly loose steering rack). Occasionally, if just one shock is back (and the other is good), some pulling can be felt.
Bounce the car on each side...see if it's about evenly "bouncy".
"Eyeball" the front wheels from straight ahead of each wheel....do both front wheels look about vertical (or is one tipped in at the top more than the other, indicating something's worn). Check for a cracked lower A-arm...not uncommon on Midgets.
Jack each wheel off ground and shake the wheel at 12 & 6 O'clock.....then shake it at 3 & 9 O'clock.Loose the first way could be king pins. Loose the second way could be steering rack/tie rods. Loose both ways could be wheel bearings.
Loose rear end could cause "steering effect" during acceleration and braking. Not really very common in Midgets. But if nothing in front is amiss, lift the entire rear of the car off the ground (not by lifting it on the rear axle) and see if the axle seems loose , especially front to rear. (might be worn spring bushings, loose U-bolts, loose shakle bolts).
With car sitting on level ground, you might try using a tape measure to see if the distance from the front wheel to the back wheel (center to center) is equal on both sides. If unequal, *something's* got to be wrong.
These are just general ideas....Obviously a 23 year-old car can have lots of little things wrong that can lead to this type of problem.
Good luck.