Okay - so this is not intended to be snarky - so - please read as a trying to be helpful idea.
A family of 6 with little ones. Even with shoes on for those under twelve - that's tough through airport security. Sweaters to take off, pockets to empty, shoes off for adults, carry on items on the belt, watching your kids if you get separated, etc - it all adds up.
Do you really NEED four separate bags with liquids in your carry-ons? Airports sell stuff, hotels sell stuff, and you are checking bags. Yes - I understand if there are liquid meds that you need some of in case luggage gets lost, but really, what else are you taking?
For travelers not checking a bag or in need of liquid meds, I get why those folks put their 3-1-1 quart bag of liquids in their carry-ons.
Can you shrink that to one bag with just the stuff you need in case of emergency during the flight? You'll have a much smoother start to the trip with less to deal with at security.
And also - if you have to check ANY bags at all - how many carry-ons do you need? You already have to deal with collecting a bag at baggage claim - so that time is lost regardless. I realize checked bags might cost you money - but there is also the benefit of less stuff to keep track of through security, waiting at the gate, and trying to find room in the overhead bins. Sure, an older kid can be completely responsible for their backpack, maybe. Are yours? Or would it be easier for you to have fewer carry-ons that might be communal?
Only you can know what is best for you and your family, but really - I've seen families struggle with WAY too much stuff in the airport and I've seen clearly seasoned traveling families who breeze through with nothing but a single backpack (usually carried by dad). Just figure out who you are.