I mostly agree with you, but you're leaving out the critical factor, which is the motivation for the people who stay behind on earth to develop and fund this project. The US got a return on the investment of going to the moon: USA #1 FUCK COMMIES. If we could establish a moon or mars base without breaking the bank it would be similarly useful. Commercialization of anything beyond satellites seems a long way off, but there are at least ideas in development about asteroid mining although the engineering challenges are still daunting. Everybody loves to talk about the trillions of tons of platinum in them thar asteroids, but they don't mention that minerals on earth have been concentrated by eons of geological, hydrological, and biological processes that didn't take place on a dead asteroid. There's tons of platinum! There's an atom over there and an atom over there and an atom over there...
But as far as launching an interstellar colony mission that will take 10k years (earth frame of reference), those people will have (figuratively) fallen out of the light cone of human civilization. Might as well have been swallowed by a black hole. No one will hear back from them for 20k years. No resources will ever be traded between the homeworld and the colony. There is no good PR, there is no ROI, just vast resources shot into space to be forgotten about after the next celebrity scandal knocks the launch out of the news cycle.
I imagine launching such a mission would be one of the most expensive endeavors in human history. And there is absolutely nothing in it for anyone who stays behind on earth, not even the satisfaction of "well at least somewhere else humans have survived if we all die" because everyone on earth who had any part in making the mission happen will have been dead for 10k years before the colonists arrive and 20k years before their descendents (twice as far removed from us as we are from the cavemen) could hear back.
Any attempt to fund such a mission will be met with very obvious responses: "that sounds neat, but what's in it for us? And don't we have plenty of other problems to which we could direct these resources?"
You would have to make it cheap enough that nobody cares that some nutjobs are taking off to the far corners of the galaxy on their own dime. And that ain't happening without new physics.