>Lack of selection

This is a common misconception. People think natural selection works like this: Individuals which don't survive to reproduce are selected against, and individual which do survive to reproduce are selected for.

In that case, you would expect an expanding population to be experiencing little selection, because most individuals are surviving to reproduce, right?

But it's not correct, because what natural selection actually selects for is individuals who reproduce the most. There aren't just two buckets, where an organism is either in or out. If a trait results in the production of, say, 10% more offspring on average, selection will favor it.

In a growing population, traits which enhance reproduction on average will spread, and that's natural selection. And it will happen more efficiently too. In small populations, natural selection is countered by drift. Basically, a trait that's beneficial might not be selected for, because the individuals who have that trait might happen not to survive for some other reason. Chance might just be against them. But a big population reduces the effect of chance (and therefore drift) for the same reason that it's much easier to roll a 1 on one dice than on ten dice all at the same time. And as you note, a larger gene pool also means more variation to draw potentially beneficial mutations from in the first place.

> means many unfit genes propagate

Also, "unfit" genes don't exactly propagate in the absence of selection. Take a gene for something (say, a medical condition) that was previously harmful, and remove all selection on it (properly speaking this means the gene isn't unfit anymore, but we'll disregard that because I know what you mean). This won't actually cause the gene to propagate, instead it's expected to remain at the same frequency in the population. To actually spread, there would have to be some active benefit of having the version of the gene which (formerly) caused the disease. You could in theory have a slow growth of gene frequency due to mutations, but this is a slooooow process.

And remember, if the medical treatment allows people with the gene to survive, but they still have reduced fertility or are otherwise less likely to have kids than the average person, that gene will still be selected against.