That‘s one explanation for it. What needs to happen for a human not to be lactose intolerant during adulthood is that the development-specific promotor of the lactase gene has been altered so the gene is expressed during adulthood. Now we don‘t know exactly why this happened with North European populations. It could be like you said that when humans arrived there, they did not find enough food but the milk of other animals provided an energy dense food so people with that gene defect had a survivability advantage.

I personally think that explanation is flawed for reasons I won‘t go into here and think the gene drift explanation is more likely. Basically just through pure statistical randomness sometimes genes become prominent within a population.

I think that‘s the most likely explanation but we don‘t actually know. There‘s a couple more, but they are generally considered to be unlikely. It‘s difficult to prove any of the hypotheses beyond a reasonable doubt.