So far, the only thing that was cleared up is what the pins are for, not why your machine is off-balance. The metal plates are supposed to slide around a bit to maintain balance, however there are 3 friction pads placed on the mating surface of the support plates which retard excessive tub movement. On older machines, these can wear down, requiring replacement.
Hopefully, you did not lay this machine on its side to do your repairs or you likely over stretched the tub support springs.