It's not irrelevant. But it's also not the only factor. Other factors include what you're looking for, whether it's quantitative, semiquantitative, or qualititative, and also the prevalence of the what you're looking for.
For example, if your study question is just whether a single user dropped out of Apple Music, and the prevalence of the drop out were 99.9%, you wouldn't need a big sample size at all. Probably 10 users would be sufficient to detect whether or not there's a single user who dropped it. Now say, the prevalence is 0.1%, you'd need a MUCH bigger sample size to detect find that single user who dropped. If you only surveyed 10 users like before, the chances are high that you wouldn't find a person and therefore concluded that not a single user dropped Apple Music, which would be a false negative.
Statistical power analysis is about calculating what your minimum sample size needs to be, in order to be fairly confident about your results. You can never be 100% confident unless you have 100% of the data. Which Apple does have, and MusicWatch does not.
To your first point though, population size is relevant. Let's take another extreme example. Let's say you have a group of 10 people, and let's say your study question is again whether or not a single one of them stopped using AM. If you selected 5 people, and none of them dropped AM, would you conclude that no one stopped using AM? No, you'd survey all 10, right? The conclusion becomes even less reliable if your asking what % of people dropped AM. After 5 people, if one of them dropped it, could you reliably conclude 20%?
Now say you surveyed 1000 people. If you surveyed 500 (50%) of them and found a rate of 20%, you'd be more confident because your sampling is bigger. And unless your selection of 500 people were somehow biased, you could feel better about your rate of dropout. However, if you were to survey only 10 of them (1%), I don't think you'd feel as confident about your results, right? Now, just extrapolate that out to 10 million people...
And therein lies the problem. 5000 people COULD be sufficient if your sample is truly random and representative of the overall population. But it also depends on what question you're trying to answer. MusicWatch is projecting an exact %, suggesting they think that the remaining 10,995,000 people are precisely like those 5000. The problem, as I pointed out before, is that this could NOT be a randomized study. Study participation is voluntary, which automatically introduces selection bias. Therefore, the probability is very high that the study participants are not representative of the overall population of AM users. Apple's subsequent release of data showed that it wasn't.