I'd second the recommendation for stick training ... and also the idea that it IS just like working with your kids. That is of course if you are a good parent!
But also remember to see things from his perspective. If he willingly goes to the cage and gets a treat, but then is left in the cage for a while and you leave - was he rewarded, or punished?
Reward/punishment is not about what you intended, but about what he experiences.
In addition to stick training, I'd recommend teaching him to go in his cage. To do this you do not just reward him when you get him in the cage, then go about your day (picking up the kids, etc). Instead you need to devote some time - not much, a couple minutes a day will do - to *teaching* the go-to-cage activity.
First you have to find a reliable way to get him to go into the cage. Hopefully a stick/perch could work, or if necessary toweling/grabbing, but I suspect a little bit of patience and prior planning so you are not in a rush to get him in will make this easier. Ideally you'd want to *lure* him in the cage with a favorite treat - but you may have to work up to that if he's already formed negative associations with the cage.
In any case, when he gets in the cage, give him a favored treat (through the cage, him in, you out) and give him lots of praise - then IMMEDIATELY let him out of the cage again and let him go back to what he was doing.
You want him to learn that going into the cage gets him ONLY good things and he doesn't lose all the fun things he was doing outside the cage. At least that's most of the time. If this is the case most of the time, he'll be okay with accepting the other times when going in the cage means he just gets a treat then gets left there for a little while.
As odd as it sounds such variable reinforcement schedules (when they get rewarded 'randomly' for doing the right behavior rather than consistently) is *more* effective than continuous reward schedules.
As you work on this, over time gradually extend the time in the cage from an immediate release, to him staying in for a couple of seconds, to several seconds, to a couple minutes. You want him also to learn that even if he gets put in the cage, you'll get him out soon enough.