One of my favorite takes on this concept was Poul Anderson's Tau Zero [1], which used a Bussard ramjet [2]. Apparently, in the 70s, in was thought that there was enough hydrogen surrounding our solar system to support interstellar travel.
"I sure as hell can. Once a crisis is past, once people can manage for themselves ... what better can a king do for them than take off his crown?"
https://en.wikipedia.org/wiki/Bussard_ramjet
More recent thinking on the concept has centered around magsails which turn the drag into a good thing. Decelerating a starship is an even tougher problem than accelerating one, and magsails are a great choice for that. (And might even be able to get a speed of 0.2% of light for departure on the solar wind)
Remember, drag is a function of relative speeds. A hypothetical example with zero velocity would allow you to gather fuel without any drag.
Now for a very large and 'slow' generation ship you need a lot of energy to keep the crew alive, able to manufacture repair parts, keep the lights on etc. Now, say you want need 1 ounce of fuel per hour that does not seem bad but if your talking a 100,000+ year trip that's 54+ million pounds.
Sure, that kind of trip does not seem appealing, but remember taking 4x the mass at 1/2 the speed takes the same energy. Further you are going to want to bootstrap a civilization at the other end which means outside of grey goo taking a lot of stuff. With the added benefit of being able to go somewhere else.
PS: You also get more energy from hydrogen the further up the chain you go. A multi stage reactor that's spitting out lead provides more energy.
What do you mean by this, zero velocity within an atmosphere won't gather anything?
Rest of the idea:
The Ramjet works by collecting hydrogen and Helium from a large area because you have a high relative velocity to the medium which also imposes drag. Think filter feeding whales. So you are collecting linearly more matter and thus energy per unit time with increased speed. However, drag is a function of matter collected AND relative speed so something like velocity ^3.
This suggests there is some point where you get less energy from collecting that you lose in drag. But, this also means below some speed you get more.
if you expect to take a 100,000 year trip you should expect to live off the land and mine Kuiper belt objects and rouge planets. And figure that once people have lived 10,000 years under those conditions they probably won't find anything interesting about terrestrial planets.