First, while the ski resorts you list have good ski schools, if you want the most beginner/family friendly area, go to Beaver Creek. Lots of greens and blues, great instructors, and the lift attendants even brush the snow off the seats for you.
If you are really concerned about your kids on lifts, you'll need to hire a private instructor for your kids. Alternatively, you can shadow your child's class all day and ride the lifts with them. The instructor may not be pleased with you as it may be distracting to your child and the other kids to have you tagging along all day. Hint, if you go this route, tip well. $$$
At 5 and 6, your kids will be handed off to random adults to ride the lifts (unless perhaps you tip the ski instructor at the beginning of the day $$$$ to ride only with your kid, and even then, I wouldn't guarantee it will happen). The resorts that you are considering have beginner lifts but your kids will likely graduate from those areas in a few days. They will move on to lifts that serve green and blue terrain. Those lift attendants are accustomed to loading ski schoolers.
At 5 and 6, your kids are not toddlers. They are capable of knowing if they are fully on the seat, and if they need help. Teach them to ask for help if they need it.
One of my sons fell off a lift at Deer Valley during ski school when he was about 7 (approx. 5 years ago). I can't recall the name of the lift but it serves mostly black terrain on the far left edge of the resort if you are looking at the trail map. Son was in a class with mostly 10 year olds so he was the short one. The lift attendant was accustomed to skiers who can ski blacks being tall enough to get on the lift. This lift was just a few inches higher than any other lift. Son was a tad too short to get his bottom fully on the seat. He knew he wasn't quite on the seat, so he slipped off to the ground at the end of the lift loading area (about 6 feet off the ground). He was fine. At the end of the day, the instructor is raving to us about how son outskies the 10 year olds. As we start to walk away, instructor says "Oh yeah, by the way, your son fell off a chair lift today." Scary for me as mom? You bet. Did I keep him out of ski school after that day? No. The terrain he skies scares me more than the lifts he rides.