Moving beyond the idea of capping, what role should parties play? How can we establish institutions to make them more a part of the game?
Depends greatly on your overall goals.
The easiest way to make parties more influential is to adopt electoral systems that parties can use, namely proportional representation, whether that be by party list or keeping STV entirely. I'm not personally sure where I stand on parties. Obviously, everyone should have the right to form any party they wish with any number of members for any purpose, and it seems quite absurd to suggest otherwise. The only question here is whether we should adopt a constitution that specifically gives parties some sort of power. Take the German electoral system, for example, as the number of seats in parliament are directly influenced by the number of party list votes each party receives. Even if an independent were to win a district, that would have virtually no effect on the balance of power in parliament. In the real world, it's not a system I'm very sympathetic to, but Atlasia isn't quite the same thing.
Parties in Atlasia have lots of potential. In a parliamentary system with some sort of proportional representation, it would be very interesting to watch parties have to form coalitions in order to reach a governing majority, however the growing influence of parties might alienate several members that do not wish to pledge allegiance to any specific political group.
Overall, I'm not entirely sure where I stand on parties, but they're certainly something that we need to consider in our new constitution.
