I play Sins of a Solar Empire vanilla (that is, no Diplomacy / Entrenchment), and usually play Advent.
Judging from the posts on this forum it seems colony capital ships are generally the ships most players start with. I used one myself a lot, but after trying out other ships first I don't get why anymore. So what are the advantages of using a colony capital ship first? Although the colony capital ships - especially the Progenitor - have good support abilities, if that is all they're good for then carrier capital ships in particular would be more useful, compensating with sheer superior damage. Clearly the main advantage has to be economical, where you save ~800g from the colony frigates you no longer need to build, and then maybe 2k gold from the cheaper upgrades. Against this you are considerably weaker when you finally run into your opponent, which means you lose more ships / position / buildings, which can easily offset the gains.
Now if using a colony capital ships means you can colonize planets that you normally wouldn't have been the first to reach otherwise, I can definitely see the point of using one. Problem is, in my games this has generally not happened. I'd be able to colonize about 3 planets / asteroids, 4 planets / asteroids at most. I can easily compensate for such a low number by building colony frigates. Sure, the colony frigates regenerate antimatter slower, but I can also compensate for that by simply building more colony frigates (they are all built with maximum antimatter, after all). After colonizing, I can scuttle them to recoup some of the losses, or if there are neutral extractors around I can just camp the colony ships there. You might save 3k gold now, but later if you lose (say) 2 Aeria Drone Hosts, that's all your advantage gone. A carrier capital ship with 7 squadrons in play can easily save those two Aeria Drone Hosts simply by having more damage available.
What else is there? You can colonize while fighting the militia with a colony capital ship, which is a good plus. On the other hand, if you use something else, you can clear the militia faster as well as colonize heavily-guarded planets (desert, terran) earlier in the game. If you take out your opponent's planets later in the match, you'll be able to colonize them at once with a colony capital ship. On the other hand, you need to be able to take out the planet in the first place, when some other capital ship probably serves better.
The advantage gained by using a colony capital ship is temporary, while that gained by using something else is more or less permanent - that something else will gain more experience, reach level 6 faster and generally stay alive the whole match. Some part of that advantage is washed away when you get a second capital ship, but for me at least that takes quite a while (maybe 20 minutes into the game) and the second capital ship will be lower level.
I've generally had better results when I started with the Halcyon compared to the Progenitor, and my friends have had better results starting with Kortuls and Marzas compared to Jarassuls and Akkans.
So I have to ask ... what's with colony capital ships, and why are they so vaunted?