The problem with it happening on the attacker's turn is that you'd have a save against the ongoing before it ever did anything, which would be more problematic.
And, yes, it's probably not worth house ruling for most groups. That doesn't necessarily answer whether it would be better or worse than the core rules, though... because honestly I'd rather figure that out, first.
That is to say, if the position were reversed and the core rule were to apply it before you saved at the end of the round, would moving it to the start of the turn be a benefit worth the change? Probably not. But it might be better or worse, nonetheless.
I think the basic logic was as follows (using an example):
(assuming the PC had the better initiative)
Round 1, the burning skeleton hits you and you are on fire.
Round 2, in the PCs turn he first burns for 5 points, does his actions, gets a save, does or doesn't put out the fire.
Subsequent rounds this continues. If at any point the ongoing damage KILLS the pc, that represents him burning to death at SOME point in the round.
Should the skeleton go first this reveals the first (minor) flaw in this theory since now the pc takes damage before a whole round has gone by, and may even die without ever getting a save.
What I was suggesting is that the ongoing damage happen at the start of the skeleton's turn, so it happens once per round exactly one round after the effect was initiated. YES, this does mean you would get a save before any damage is inflicted, which is a pretty minor decrease in the effectiveness of ongoing damage. You could always offset this by doing the damage at the END of the skeleton's turn, in which case everything is pretty much exactly as before!
I think that answers the basic rules mechanical questions. It will work and depending on exact details things come out either identical or similar to the standard way, except it seems a bit more logical when the damage is done. OTOH from a playability perspective I wouldn't bother to change the rule, it is simpler to do the damage on the PCs turn, and it would be my guess that was what WOTC thought too.
For myself, I don't think it is worth 'fixing' or especially broken. I was just making a few observations.