i think the turning on aang in bato of the watertribe is more complicated than that as katara and sokka have been waiting to hear from their father for so long but they also understand why aang did it (given that they do indeed want to go meet with their dad asap/aang doesn't have anyone else, he only has them),
but them deciding to stay with aang also really cements them as family too now (like katara has said it before but there's a difference between saying it, the urgency created by winter solstice, and the later stages of ok well would literally die for you)