Let me start by saying that the pressure has currently way too high impact and it should be greatly reduced. I've just started playing GC 3, but this was the first thing that struck me as unbalanced. Looking in the forum I see that I'm not alone in this opinion so I expect that this would be addressed through beta changes.

However, pop pressure could be used to open door to add another aspect in the game (assuming this doesn't already exist in game): migration of population to other planets. First, migrating populous to other worlds would somewhat rebalance the negative/positive effect under the same domain. The majority of migration would be among the neighboring planets so the impact is not instantaneous. The planets with insufficient population would grow while with overpopulated would also benefit by reducing population that it cannot feed.

Migration and lack of ship availability is another story, but this can be also addressed and potentially be used to build on this game aspect. For example, migration can be done be reusing existing scout/explorer ships to do planet-sponsored migration. Spending time on migration instead of exploring would need to be balanced against the impact of overpopulation. Maybe a new pod needs to be added to ship design to enable this?

There could be also aspects of illegal immigration, ships-rafts, etc.

The logic also follows the migration of people we see between countries: people migrating from overpopulated and poorer countries to economically stronger ones