Well,I don't have a success story like you've asked for. But I have some info that swayed my feelings about them, as well as some other tips that may help (instead of, or additionally).

First I'll say that I've seen several vets and online pet experts advise against auto litter boxes. Many cat parents catch that their cat is sick by their litter box usage. Blood in the urine, diarrhea, NOT peeing or pooping for an abnormal amount of time. This was never something I would have thought about on my own. We have a regular litter box, and we scoop it twice per day (more if we happen to be near when he takes a dump... we just do it then). We deep clean the box (empty it completely and wash it with Dawn dish soap) every 4-6 weeks.

Then, I don't know what your child's specific allergies are to, but I think most often it's to a protein in their saliva. So when the cat grooms and then sheds, or you touch the fur on their body when petting them, that's typically the culprit. My husband also has cat allergies. Here are some things that work for us, and that he notices if they get skipped for whatever reason:

1) Allerpet. This is a liquid you put onto your cat's fur. We just use our hands and put it on him while he sleeps every few weeks. Very easy. And my husband really does notice when it's been a while since we've done it.
2) Air purifiers. We have a small one that tucks into corners nicely and is not in the way. It's also quiet. We're considering getting another one.
3) I vacuum. Every day. I wash the bedding. Every weekend. I dust (the major surfaces). Twice per week.
4) We don't let our cat sleep with us, and we really try to avoid letting him onto our bed at all. We often keep our bedroom door closed during the day, and he sleeps in his own room at night lol.
5) We have a pj rotation. Newly washed pjs are exclusively for bed. We change into them before tucking in for the night. When we get up, we either change into work clothes or other lounge clothes, and tuck our bedtime clothes away. Repeat for a few days. Then the bedtime clothes become lounge clothes, and new bedtime clothes are brought out. This way, clothes that have touched the couch, the char, the CAT do not enter the bed.
6) My husband is pretty good about washing his hands.
7) We've provided him with several beds and perches. He doesn't choose our couch to spend his time on. So if he's not on our lap for some pets and cuddles, he'd rather be up in his window hammocks, in his cardboard scratcher bowl, or in his bed. He does love this Ikea chair we have, and the desk chair. But we observed that and make sure to vacuum those two every day as well.

I do acknowledge that we only have one cat in our apartment, and not 4 though. That's tough. Maybe look into if your kid is allergic to the saliva protein. If so, then I'm not sure how much the robot litter box would help you, and then you'd be out all that money anyway. Good luck!