[DanBC]

> In live, non-corporate language would any native speaker in fact routinely use "they" instead of "he" when referring to a user?

Yes, lots of people choose to use words like "the user" rather than "he". Or they use "he" and "she" throughout the documentation. Or they'll use singular "they".

> But is it an actually big deal for native speakers?

For some people it is a big deal because they are that type of person. For other people it's not a big deal but they do it anyway, just because.