I think at that age the parents or the boy can't know if it's a phase (induced by the presence of a bigger sister, totally normal), or if he's just gay and so likes girl stuff, or if he's actually transgendered.
I think it's wrong that these parents took an input one day and then forced a binary switch of gender on a kid who's not able to understand it completely yet. If you do that, the child will go along with it, and if the parents end up being wrong, the kid will revert when it's too late and he'll already have suffered unnecessarily.
People can get confused or be insecure as to whether they are gay or trans so I don't think it's a good idea to make a decision this big based on something a THREE YEARS OLD kid said. Plenty of gay people decide do switch gender later.
When I read the article and the part about the burrito came up, I thought it was something even more extreme:
https://en.wikipedia.org/wiki/Otherkin
it's not like the kid has signed into a contract that says they have to abide by their decision or anything like that. If the kid decides to reverse her decision and the parents allow her/him to do that, then what harm has been done here?
A kid is not adult. If he sees his mum is happy and encouraging him to express his feminine side (because she's totally convinced he's trans since he said it), he will go along with it, and maybe not express his desire to revert the decision. Think also of his social life. Kids like stability and routines. Or maybe he doesn't feel the issue strongly, so leaving it alone might have been better considering all the troubles as soon as the kid is in school: it will have to put up with various issues, starting from which bathroom to use, and putting someone through that if he's not actually transgendered is not useful.
IMO there's just as much potential harm in the parents forcing this kid to conform to male or female gender norms.
except the parents are doing exactly that by changing the name and doing a total switch-over to female gender instead of allowing the kid to be queer and freely find out his identity on his own, after puberty hits.
I'd just make sure he's informed that gay or trans people exist so that he has a way to find out what he is. This is the part that ultraconservative parents and societies get wrong, along with repression of expression, which leads to suicides and people going gay after they had been married or staying in the closet and married but visiting transsexual prostitutes (there's a dick but she looks like a woman so it's not gay right? I believe that's the thinking).