Actually there are a number of ways games have dealt with this in the past. Pushing through creeps could slow you down but you have to be in contact with the minion for X amount of time to start pushing it so you don't get slowed randomly when walking near creeps.
You could also use the jump through tactic as seen in games like stardew valley where pushing against a minion for a certain amount of time jumps you through to the other side. I feel like this might be a bit abusable and not clear enough/a bit random feeling in League though.
Also note that we already can push minions around, the issue is that in large groups there are more minions pushing that one back into place and as a champion pushes the minions around their ideal pathing changes. In fact most creep block is just a loop of pathing one way which shifts a minion to make a faster route the other way. The champ then backtracks and tries to walk that way but without the champ pushing the minion that other path closes up and they backtrack again to their initial pathing.
So really we may just need pathing to keep trying to force you through minions instead of trying to path you around when surrounded.