How about combining population growth and immigration but so that it's actual real immigration what's happening while also keeping it automatic and very simple?
The current +0.1/turn is a good rate for new settlements from the game-mechanics point-of-view. It gives a reasonable rate to get the production going. For small populations on new worlds it also must come mostly from immigration or you'd get a ridiculous birth rate.
Reasonable pop growth: If 1 turn is a week then let's say pop growth is +0.1%/turn (meaning +5.3%/year).
In any world that has reached its pop cap those +0.1% want to emigrate elsewhere. This is the immigration source pop.
In any world that has not reached it pop cap the +0.1% goes to local pop growth. If this is greater than 0.1 billion or greater than what's needed to reach the pop cap the remainder want to emigrate elsewhere adding to the immigration source pop. (A rare occurrence as you'd need a 100b pop world for this but let's have this case, too, as you may want to mod the turn to represent a month or even a year and thus want to increase the pop growth rate.)
Any world where +0.1% gives less than 0.1 billion and has not reached its pop cap is a target world for immigration and it can receive as many immigrants as is needed to fill up that +0.1 billion.
Then you just fill the immigration targets from the immigration source pop.
If the immigration source pop isn't enough by itself to fill up the +0.1 billion growths in all of the receiving worlds the remaining is taken from the most populous worlds who will thus lose pop to fill the new frontier worlds. To get a stable result for this mechanic you need some kind of "floor limit" rule, for example, "emigration can't drain pop below 50% of pop cap".
If the immigration source pop is more than the target worlds can receive then you have overpopulation which part is lost. They are assumed to end up living in slums outside your civilized system, emigrate to a different empire entirely, or starve. You should, of course, receive a morale penalty for creating such a dystopian future society so you better find new places for your people to live, you monster!
This way you get a limit to your pop expansion as populating your frontier will drain your center worlds.