I did change the tube.
Nothing wrong with the tire, the rim or rim tape that I could see.
The valve's stem was torn to shreds though : that's where it seems to have burst... Maybe just defective ? It was one of those tubes filled with green sealant, and some of it had gone into the valve the last time I put air in it... I don't know if that changes anything. Also, before that, I'd had trouble with the valve which seemed stuck : I'd press with the pump nozzle and it wouldn't let air in or out.


Anyways, I inflated the new tube back to near 85psi and went for a ride with the kid. No explosion so far. I'll report back if it ever does again.

Reading the posts above, I take it that I have an unusual tire size ?..
What'd be the advantage of getting a bigger wheel, as Delmarva suggests ?