About "builing the ship"
Any civilization that can build ships that can travel at close to lightspeed has other advanced technologies at its disposal - for one, advanced nanotechnology.
FOR US, building a colony ship would be a huge drain on world economy; NOT for a civilization that mastered nanotechnology and has at its disposal the resources of an entire solar system.
This is very much a chicken and egg situation. We are nearing a point where we will need to develop self-sustaining space travel, forget near the speed of light. We may eventually develop "advanced" technology, but it will be after that critical point where we either develop self-sustaining space travel or die. We MAY have already passed this point.
Since we are the only example available, one must believe that other species have gone through similar development cycles. Like I said, and you ignored, it may be a natural lifecycle issue, that by time a species realizes it must get off planet, and develops the technology to get off planet, it's too late to mount a sufficient effort.
Motivations for colonizing other places are numerous (as our own history showed) - population pressure; the desire to escape oppression; in order to start "fresh" and not be just a cog in the huge societal machine; certain cultures will wish to be "independent", etc.
The point is - any alien species will be composed of many different cultures with different motivations; some of these civilizations will have the motivation to "move out". And once colonies are established amd develop their own culture, the number of different cultures/motivations will increase (for example, since 1500 until today - aka in 500 years - humanity changed enormously/increased in numbers at an impressive rate) - some of these colonies will soon send ships further out, some not so soon. Per total, as the number of colonies increase, the number of colonizing ships will increase.
The main motivator for exploration of humanity has been lack of resources. A group wanted to seperate themselves from another group for whatever reason, so they moved to the next valley. At times when there was no next valley, simply destroying the old group was an option. But, the buttom line has always been resources.
You are likely right (and wrong) that other species will contain sub-groups with different motivations. This is the problem we have on Earth which is a major hurdle to developing the space technology we need to get off planet. Societies that are all on the same page, with no dissent, actually have a better chance at recognizing the needs of the future and acting on them.
You are counting from 1500 to today, 500 years, why? The population at 1500 was quite large, larger than a colony ship could deliver to a planet. No, we're talking starting with a few hundred people and filling up the planet, this is not a 500 year task, more like 5000 years. And, that's assuming unrestrained expansion. For a society that has spent a few generations in space will build in a certain frugality that would continue when they are society building on a new planet.
About the "dangers" and "speed" of space travel
Outer space, unlike what we see in Star Trek is BORING. REALLY BORING.
Essentially, space is a huge, empty desert with few resources of any kind (and the solar systems are "oasis"). The greatest danger in outer space is to remain without resources. The best way to minimize this danger is to travel at yout top speed - aka near light-speed - once you have decided where you want to go - and, with the FTL speed limit, you WILL NOT LEAVE WITHOUT DECIDING your destination.
Again, using the only example we have, the best we can do now is head tward a solar system we think will have a planet that we can survive on. We won't know until we get there. Of course, to get there we would have to develop colony ships that are virtually self-sustaining, so that if we get "there" and "there" isn't there, we would head off to the next likely "there". Of course, once we have self-sustaining ships, why do we need to leave the solar system any time soon? There are a lot of resources right here.
Space is not all that empty, there are rocks traveling quite fast all over that if they bump into your space ship, or vice versa, the space ship is toast. Space is a big, bad place, and a society would risk it only if it had no other choice.
Also, Near Light Speed travel will require a lot of fuel to speed up and slow down. This is a problem right now. Even IF we get to NLS, we still need a self-sufficient system for those 10s or 100s of years, which will require a big ship.
- there will BE some with the ability to create technology/the desire to expand (a definite evolutionary advantage, as said). I have been talking about these species. It only takes ONE SUCH SPECIES and a few MILLION YEARS to colonize the entire galaxy.
and it only takes TWO to foil each other.
Again, we haven't even determined if a species lifecycle allows enough time to get off-planet before it uses all it's resources. It's not looking like we will do it, what if this is the norm, or unavoidable?
We are on the cusp of manipulating our own genes. Perhaps this is a normal situation, and the undoing of every possible spacefaring species. Perhaps smart machines are the undoing of spacefaring species.
The fact that Aliens aren't here and Drakes Equation indicates they should be means there's something that makes interstellar expansion unlikely or extremely difficult. We can't just hand wave the problems away, because one of them, or all of them, or other ones we haven't even figured out yet, are preventing the Aliens from getting here, and therefore us from getting "there" someday.