My wife and I just flew with 2 kids and no checked bags. Yes, it was a pain to manage 4 carry-ons and 4 personal bags, but it was manageable, and it beat the hell out of paying another $200 to have those 4 bags checked.
Edit: They're 6 and 8 years old. It's impossible for me to say whether it would have been possible when they were younger, as my 8-year-old is only recently past her medical issues that necessitated considerable additional luggage -- so we just didn't fly.
I must admit, I recently caved and tried the Reddit app. It sucks hard.