Potty training, part 452

Sam’s not potty trained. This is hard for me to admit, especially since she’s almost 3 and a half.
From her first day in this world, Sam didn’t care if her diaper was dirty. Even now, she can smell like a cow pasture, and if I ask her, “Did you poop?” she’ll calmly say, “No. I don’t need a diaper change, thank you.”
Before I became a parent, I never expected to get an enthusiastic round of applause for peeing on the potty. Now, I’ve gotten more standing ovations for that than I can count. I’ve also gotten used to having an audience in the bathroom.
We’ve tried all the techniques experts recommend. She can tell you that, if she pees or poops on the potty, she’ll get ice cream (which is the world’s absolute greatest reward for her. Yeah, I know, don’t reward your children with food, but I’m getting desperate here). She can recite her “It’s Time to Go Potty!” book that we’ve been reading together since she was about 18 months old.
Her pediatrician swears that one out of every five 3-year-olds is not potty trained. We went down a checklist to determine toilet-training readiness, and Sam failed miserably.
I’m now of the don’t-push-her camp of thinking. I think she really wants to go on the potty, but she physically can’t control it. I’m hanging onto this line of denial until she’s 4, at which time I’m afraid I’ll have to resort to the cold-turkey diaper swipe.
What have you tried that’s worked? Do you think Sam will do it when she’s ready, or do you think I should be trying another method of training?