Being overweight is in an entirely different category than the other attributes you mentioned because it’s a choice, and more importantly because physical fitness is directly relevant to a soldier or sailor’s ability to perform their duty. In fact, the show itself addresses this: there are scenes that show Tilly jogging around the ship so that she can get a “physical endurance commendation” and be more likely to get into the Command Training Program.

If she were a civilian character, sure: no judgement, no problem. But having an overweight military officer (especially if they hadn’t acknowledged and addressed it the way they did) would be problematic in a legitimate, non-judgemental, suspension-of-disbelief-defying way.

(That said, I don’t think Tilly was anywhere near big enough for fan complaints to actually be legitimate. I’m just saying that, in principle, that category of complaint could be legitimate for that type of character, in contrast to complaints about race/sex/hair that are never valid.)