My favorite places for kids are Alta and Sundance(pics on Park City questions thread) because they both have large beginner's areas without
traffic from uphill blasting through. Both have good ski schools. Both are relitively inexpensive. Alta is a world class big mountain with a gladed flat basin in it's "lap" perfect or kids and beginners. Sundance is a small but very charming area with lots of broad low angle groomers and little woods tracks that kids love.
I've always found a flaw or two with most of the others, PCMR has a lone bunny hill with a slow lift and a loooong line. It's adjacent beginners area has high speed fools zooming through bowling children.That's my main gripe with the others too. Snowbasin is great because there are so few other skiers and lots of low angle terrain, but it's out of the way.
Deer Valley, of course, is in a class by itself. The staff is everywhere and top notch, put 'em in ski school and they don't want to leave.The bunny hill is great and has no access from above so no "bowling" effect, and unless they changed policy, you can ride the bunny lifts on Wide West for an $18 ticket. DV is becoming more like skiing in Central Park than on a big western mountain because of all the fancy log mansions and hotels everywhere. But the turkey chile is great and they do kids better than anyone.
If the free lift ticket is really important, I'd suggest Snowbird with "Big Emma". It's been a long time since I've been to Solitude, and I don't much care for Brighton- others may know better. My kids have free under seven season passes at PCMR and I've yet to take them- that's my opinion of the kids area there.(the back bowls are a different story)
I should add= this is splitting hairs- it's all pretty fantastic. If it isn't sunny, it's snowing, Another great season in Utah- that makes two back to back.