Firstly, technologically advanced species have the ability to modify their own bodies. We already do this with great success and we're barely a few generations into the era of washing our goddamned hands before doing surgery. Additionally you have to consider the difference between individuals, species, and civilizations. Can intelligent robots carry on a civilization? There doesn't seem to be any good reason why they couldn't. And once you are there, then it doesn't matter whether or not it takes 10 years, 100 years, 10,000 years, or even a million years to travel from one star to another. Intelligent, conscious beings who carry our civilization into the stars could indeed do so successfully. Even if they take many thousands of years to hop from star to star and even if they take thousands of years to build up a civilization (on a planet or in orbit of the star using resources from the stellar system) that is then in a position to send out more voyages across the stars. Even then the result will be a massive, sprawling network of our civilization spread across countless stellar systems within a time frame that's a blink of an eye in cosmological terms.

If it took 10 million years for a voyaging vessel to land, settle, grow, and send out only 2 voyaging vessels (and then stop voyaging) then they would colonize a million worlds in 100 million years. In only 400 million years they would have colonized the entire Milky Way.