So, it has become obvious that rapid colonization and expansion is the only valid early game strategy. Even with the Large Empire Penaly debuff, which limits the maximum size of the empire, but it is still the only viable strategy to reach that maximum number of planets as soon as possible. There is nothing wrong or 'off' about this. There is no reason why not to colonize habitable planets and do so quickly, however, the problem as i see it comes from the colony ships, which are OP. Now hear me out.

Strategy game should give you choices, each one having consequences, advantages and disadvantages for you to weigh out. In the early game, there is only one viable choice- colonize, FAST. Wouldn't it be more fun if you had to pace your colonization efforts, and weigh in if you want few developed colonies or many undeveloped ones ? Civ 5 has done this in a way (altho i admit i didnt like it very much). Building settlers would mean you are not developing your home town, and it would stop growth, and not be useful till later on... In GC3, colony ships are the best thing you can build in almost every scenario. Think about it... Colony ships spawn capital cities... It takes your civ around 5 weeks to create a ship that can take 3+ billion people on an untouched planet, and immediately construct a facility that can feed 8 billion people, has the power of (essentially) top tier factory, research building, and can in the very first week start to produce new ship parts. As an added bonus, it also positively affects growth from week 1.

What if we decided that this is OP for a single ship, even from a gameplay perspective (since it spawns 4 high level buildings in one tile, immediately, with no techs required)? 

First, it is very unrealistic for a single ship to be able to carry all this, along with 3+ billion people. Wouldn't it be more fun if the new  colony isn't immediately useful ?
Here is my idea how to achieve this;

1. Colony ships carry no more than 50 million pop- this, while being realistic, also limits the immediate usefullnes of colonies, as it should.

2. You do not start with a Capital City. The CC is a very strong building, that took ages on your home world to build... It is now up to colonists to build in in a much shorter period of time. What I mean by this is that you start with a settlement, that provides limited production and food generating capacity. It is however able to be upgraded to a full Capital City through social projects... OR by consuming constructors and Colony ships (which bring more population)

Think of an "expand capital level";

Level1-1  production, 250 mil max pop

Level 2- Allows social projects that are not "upgrade capital", but only 2 tiles around the capital, 2 production, 500 mil max pop(though farms are buildable)

Level 3-Allows sponsoring shipyards, 1.5 bil max pop

Level 4-3 production, Allows building anywhere on the planet, 4 bil max pop

Level 5-5 production, Removes colony LEP  8 bil max pop,

Each level would need to either consume a colony ship/constructor, or to be built like any social project

3. Civilian transport- A civilian transport ship that can take +- 500 mil pop and ferry them to colonies, but can not colonize planets on their own.

This would bring strategy to the colony rush as you have to decide whether to claim more planets early or to develop existing colonies and leave that for later. It also makes the colony rush last longer (i think this is a good thing), and enables war, with planetary invasions even before every planet is taken...