So I have a couple of kids' bikes with some seriously droopy chains at certain points of the crank rotation.

I do know most of the basics of proper chain setup. Turn the crank until the spot of tightest chain tension is found, pull the wheel back almost all the way so that the chain has just a little free movement at that point. The problem is that, after I tighten the axle nuts, and turn the pedals to a different point in the crank rotation, the chain now sags quite a lot. I know a little variation in chain tension is normal, but for a fraction of the kids' bikes I work on, the droop is very noticeable. I'm worried about a derailment if the kid hits a bump or something.

Am I right in understanding that the cause of this problem is an exaggerated case of a chainring that isn't perfectly circular? Is there any reasonably easy way to fix the issue without swapping over a crank/chainring from another bike?