Below is my initial itinerary. Due to airfare, I can spend an extra day or two somewhere. Where would you add time? We're a family of 4 with kids age 12 and 10 who enjoy the outdoors.

Day 1 - Fly to Denver (CO Springs)
2 - Colorado Springs (CO Springs)
3 - Colorado Springs (CO Springs)
4- Drive to Durango via Great Sand Dunes (Cortez)
5 - Mesa Verde (Cortez)
6 - Four Corners (Cortez)
7- Drive Million $$ Highway (Montrose)
8 - See Black Canyon of Gunnison, drive to Estes Park (Estes Park)
9 - RMNP (Estes Park)
10 - RMNP (Estes Park)
11-13 Drive to South Dakota, see Mt. Rushmore, Badlands, etc.
14 - Drive to Denver
15 - Fly home

Lot of things to do/see in Mesa Verde area (canyon of the ancients). If you plan on going to the four corners monument, it's not worth it, unless you like middle of nowhere pointless jaunts.

Wow, you're at about 1,800 miles on this trip. Looks exhausting! Skip South Dakota for sure. Spend more time at the spots you have-- another day at Mesa Verde and Estes Park. Or If you were to add another side trip/spur on your loop, consider a shorter detour to Aspen (Maroon Bells) or to Breckenridge. And even that might be too much. Hopefully Fmpden will see this post. He is the one who has helped me in the past. Also read some trip reports on Mesa Verde on everything there is to do there.

OH wait. You are driving BACK to Denver to depart. Definitely save that visit for another time.
Spend a few hours at Woodland Park about 20 miles from the Springs to see their dinosaur exhibit.
Another day at Mesa Verde as suggested. That is a long way to go to shortchange it.
The kids will enjoy spending a few hours at the Great Sand Dunes (IMO)
Make reservations IN Mesa Verde at Far View Lodge instead of staying outside the Park.
Agree about Four Corners.
Spend some time between Gunnison and Estes Park. The highway at Glenwood Springs has an exhibit about how it was built--don't miss that. Maybe stay in Glenwood Springs and enjoy the hot springs.
At Copper Mountain is Woodward --a facility where trick skiers and snowboarders practice their routine by going into huge foam rubber pits. Your kids can enjoy jumping around on trampolines into the pits.
There are places with zip lines--maybe Breckenridge. More than enough to fill your nice length of time with great things.

Before kids, DH and I traveled 4600 miles in 17 days (Seattle - Banff - Glacier NP - Yellowstone - Seattle). We did not see everything but got an idea of where we wanted to go back.

We are doing SD this trip because the kids (and I) really want to see Mt. Rushmore and it makes the most sense this trip.

I know we won't see everything but I don't really expect to.

Spend your extra two days on the Mount Rushmore leg. Make sure to carve out time to see Wind Cave, which is a separate park from Mt. Rushmore and the Badlands. You could also spend a day in Deadwood. There are so many things to do in the area, I can't imagine you'll have trouble filling up the additional days.

Also consider adding Royal Gorge to your itinerary. Since the Pikes Peak cog railway is down, I'm not sure it's worth including that when you're in Colorado Springs. It's a punishing, stressful, traffic-filled drive, so this might make a suitable replacement even though it's pretty touristy.

I'd also skip Four Corners and add extra time in Mesa Verde. Despite all the driving, this sounds like a great trip to me. I'd have enjoyed this as a kid.

I would pull a day from Colorado Springs, add one to Mesa Verde and one or two to South Dakota.

Hands down add time to South Dakota. Search under my name and find the trip report I did for Yellowstone last year - the last part is the South Dakota portion of our trip. I was stunned by how much we enjoyed that part of the country. With kids your age definitely go to Mammoth Digsite - that's at least a half day trip. Also if at all possible get tour tickets for Minuteman Missile Silo - we did not plan ahead and so were able to do just the visitor center/look at missile but there's a tour which takes you into the bunker. Finally, visit Rushmore not just in the day time but go to the night "flag lowering" ceremony. Very moving and patriotic. Have a great time!!!

Definitely don't go up Pike's Peak--even if the cog railway was up, IMO. If you want a mountain view, drive up Mt. Evans when coming back toward Denver.
The Badlands is one of the most interesting areas of the west --to me, moreso than Mt. Rushmore especially for kids.
BUT I still say there is so much in Colorado that you will "drive by" and not experience by extending your trip to SD. It's basically a 2 day round trip drive just to get there.
Mesa Verde is a true wonder of the world and as I suggested, stay in the Park for 2 days--the drive in is very long.

Not to pile on, but I too agree, avoid 4 corners and add time in Mesa Verde. Do as many of the ranger led trips to the dwellings as you can, the kids will get a charge out of climbing up those ladders.