= Method
Load Control group's save, i.e. no blockers cleared, no districts built on Earth, Archives is kept.
Take Expansion tradition, pick Colonisation Fever (but no further traditions), do Superior Colonies council agenda.
Survey a continental planet (90% habitability: humans are Adaptable), send colony ship there, `fast_forward 500` (the `populate` command froze my game), save game when colony is set up.
Start counting months from 2206.01.06 when the colony came online.
== No Resettlement, No Construction
Build nothing on either planet: homeworld and colony make no districts, no buildings. No new jobs on the colony beyond the starting 200.
View attachment 1295086
Ok, this was a useful experiment. Civilian positions don't count as jobs when pops consider immigration. Civilians aren't going to spend their own money to travel to another planet just to do the same kind of civilian jobs, they want government-associated jobs if they're going to make that move. Also good to see it takes about a year for 100 pops to naturally immigrate (at least for UNE humans with Nomadic).
== No Resettlement, Colony Construction
Load save, this time build 3 resource districts (mining, generator, agriculture) on the colony. Still no new districts on Earth. Natural build times, no console command.
View attachment 1295094
View attachment 1295095
Nothing out of the ordinary, when there's more jobs to fill, more pops migrate to the colony. With more living room, the total population of this 2-planet empire is better off than the undeveloped colony.
Also worth noting: Earth had 5400 pops at the start of this save, which from my OP is around what they had by that save's date. Starting a new colony is still conjuring pops out of thin air, just that the Expansion tradition no longer has an option to conjure twice as many in a new colony.
=== Resettle, No Construction
Load save, change law to allow resettlement, move 900 civilian pops from Earth to colony. As before, no construction on either colony or Earth.
View attachment 1295099
For total population, it's better to have done the resettlement even if there's no immigration than to slowly wait for pops to naturally migrate. Plotting the change in total population per month:
View attachment 1295100
Each dataset's mean is also drawn as a horizontal line.
Now, how useful is the in-game tooltip? Below I've plotted (for the colony) the "estimated growth change" shown in the management tab above all pop groups, i.e. the predicted total change:
View attachment 1295108
In the Natural Immigration case, the estimates don't track immigration at all. In the Seeded Resettlement case where there is no immigration, the estimate is a reasonable prediction for how the RNG will do. Here's the same graph for Earth's tooltips:
View attachment 1295116
Here, the Seeded Resettlement case has the tooltip frequently underestimating how many pops will grow from the RNG.
= Conclusions
I see nothing that refutes the prevailing theory. In realistic gameplay, you'd be building a luxury apartment for the initial amenity boost on new colonies and you'd be building some resource districts so you'll easily have enough capacity and jobs to not be throttled on those factors. It's the raw pop numbers that matter most in a new colony.
According to the theory, if you have a planet at say, 2400 population, you want 7200 real capacity to get a solid base growth rate of 1+3 (as the tooltip will say). That means 4800 to cover with unused housing and unused district slots. An unused district slot is worth 400 houses in real capacity (assuming a standard planet). So 12 unused + city districts and aiming for 2400 population is a good rule of thumb to aim for.
If you have a 4000 population planet, you want 8000 real capacity for the same 1+3 base growth rate. That's 10 unused city districts, not much different from the 2400 pop case. So the "12 unused + city districts" (or 8 if you have a luxury apartment) is a simple number to aim for.
Now what does the theory say about wide, thinly populated planets vs tall ones? At the same real capacity, a 300 pop planet has base growth 1-0.28=0.72, a 600 pop planet has base growth 1+0.38 while the 2400 pop planet has 1+3. 8*0.72 > 4 so technically having those 2400 pops spread across 8 planets is better than as 1, but this is impractical in the earlygame: better to have tall planets with medical workers benefiting more pops than building medical facilities 8 times and dedicating so many more workers for the same growth bonus. It's another matter when you have robot assembly and clone vats online but the start should be tall planets, not wide. Whether your borders should be wide is another topic entirely, this is just about how many planets to rush colonising in the earlygame.