I'm guessing they insisted for the kids because if your kids were sent back, the airline has to front 4 seats since obviously your kids won't be repatriated alone.
Speaking from my personal experience, if you don't cancel your extension before leaving, it WILL be a pain in the arse to get a new one when you return.
I have found it takes on average of 35-50 mins, depending on how many people are ahead of you. Pro tip: print the form online + bring your own picture to save 100-200 bht (single vs multi permit).
Another thing, Thai immigration can only grant work visas to teachers with a bachelor's degree. Without a degree, they cannot. This is for teaching. If you wanted to do other jobs, they CAN grant you a visa without a degree.