stronger wheels, larger tires, rack, and a trunk bag(that sits on the rack instead of hanging from it on the side).
this will fix all of your problems.
honestly i have two bikes that can fit a rack, but one's a fixed gear that i don't want to use for commuting, and one's a mountain bike that i'd have to completely neuter in order to commute on it quickly.
because of this i've been commuting on the mountain bike with the pannier or using the road bike and a backpack converter for the pannier. honestly i totally hear you about controlling the weight over bumps, but that's really just code for saying "my commuter bike is too lightweight and weak."