Has it ever been impossible to evacuate planets within days? I mean, we have evidence of colonies being evacuated in far less ("For the Uniform", say) if the folks have ships of their own, and of shipless colonies in general being very, very small and easily beamed up by a single starship. In turn, things that stop transporters would normally tend to stop colonization, too, while a colony 15,000 people strong would be quite unlikely not to sport ships of its own.
And it seems inevitable that better-established, non-trivially evacuable worlds with hundreds of thousands of people are covered by different rules altogether - the whole situation only arose because there were illegal squatters on Sheliak property, not because there would be an established UFP settlement there, and those normally tend to be separate things.
The fun thing is, evacuation is hindered by lack of surface-to-orbit transportation; apparently, there's no problem in accommodating 15,000 people aboard the Enterprise
as such. And while all Starfleet technology seems to go haywire, including sensors, transporters and weapons, our heroes seem to accept that Sheliak sensors work (they saw the colonists) and Sheliak weapons work (they are a deadly threat to the colonists). So the thing Picard should be asking for is the use of Sheliak transporters in moving the colonists from Tau Cygna V to the Enterprise
Why, then, does Picard not negotiate for this from the very start? Why does he at first tell the Sheliak he doesn't want to uproot the colonists? No wonder the Sheliak find Picard's attempts offensive.