Two of my favorite space games are Mass Effect and Star Control 2. Both of them get reflected in Stellaris - but there's one thing missing.

In both games, you could fly around the galaxy and interact with other species, helping or hurting them through your actions.

Right now there's an issue that anomalies tend to be cut off when you encounter alien empires, leaving you with few events in the mid game. I think that void could be filled with events involving those empires! What if an alien empire had a sun going nova? You could fix the problem and win their gratitude, or accept their refugees in your empire, or even sabotage their efforts to solve the problem, making the damage worse. What if you caught an alien ambassador involved in a plot against your leader? You could expel him, execute him or drug him and send him back with explosives. What if the robots on an alien world malfunctioned? You could figure out the cause and help them, or offer to buy the broken bots at a fraction of their value, then fix them yourself (but if they find out, they'll protest and your relations will go down). Or maybe you could try and fix the relationship between two empires? Perhaps an ancient artifact with religious value was stolen from one empire in the last war and you could negotiate its release? There's so many possibilities!