I had always thought that the fundamental forces were largely the same regardless of whether time was reversed or not.

Not really. Even the electric force is not purely time symmetric - you have to flip the sign of the charge if you want to flip the direction between forwards vs backward in time.

Even worse, the weak force breaks another symmetry as well, parity symmetry (which basically means that moving backward in time, weak force particles "look" like their mirror image, instead of looking the same).

Theoretically this holds true, but in practice it never happens.

Why is a major question, but any understanding of our universe must assume this fact.