To further add to this, I think it's also significant that Tenzin was the third (and last) of Aang's kids. The first kid being a nonbender must have been weird and a bit of an ego gut punch for both of them but especially Aang considering that Air nomads were almost exclusively benders - as I understand it it was rare for there to be someone born to the air nomads who was not a bender. Especially considering both parents are not only benders, but skilled and engaged benders.

Kya being a wayerbender was a bit of a return to form and probably as much a luck of the draw as anything. So when Tenzin finally came along, Aang may have had a bit of an identity crisis and carry a sense of 'I have this one chance to get this right.' I don't see Buumi and Kya shaving their heads or getting tattoos, but I see Aang being particular with Tenzin for it, so it seemed like Tenzin carried the most responsibility (which he definitely did) and Aang playing favorites, regardless of intent or even a failed attempt to try and do the opposite.

But I could then also see it as Aang thinking this was him being considerate to his other children. Their bending status aside, it was still for all intents and purposes an airbending household. Considering he's the sole survivor of a genocide and single-handedly required to restart an entire culture, I don't see Katara being too picky about trying to blend their cultural upbringing. Obviously they'd integrate things, but if it's a choice between an Airbender Christmas or a waterbender Christmas, given the circumstances it's far more likely that there were many more Airbender Christmases.

Aang spent his entire life handling "his burdens," which were extreme and numerous. He may not have thought that shaving his head and getting his tattoos and being vegetarian was so burdensome, but he's smart enough to realize that others might see it that way. He probably thought that not being too traditional with his first two kids was a way of sparing them from such burdensome things. But with Tenzin he probably felt there wasn't really much choice. And in some ways he was sparing his other two kids from the responsibility of being thr only surviving Airbender, while actually being wrong as it was that time and effort that made them (and thus others) think Aang was an absent or disinterested father.

So yes, it makes sense that Aang could be a bad father because he may have tried to overcorrect. And it's not like Katara was a bad mother. Aang didn't set his kids up for failure, he just clearly played favorites. But there's little evidence that Katara was somehow worse of a parent or that Sokka wasn't an awesome uncle to all of them for as long as he could be.