If a chain were worn to the point of skipping on cog teeth having the same usage, I would be surprised if a new chain on the old cogs was not actually much worse.

With fresh chain on worn cogs, the problem isn't as it first seems. Rather than the rollers being pulled past worn teeth under increased force from pedaling, what is really occurring is that the rollers fail to drop between the teeth because of the hooked corner striking the roller.
The newer chain runs in a relatively "advanced" position on a worn cog, since the driven surface at the base of the tooth has worn away. But since the hooked tip of the tooth has not been worn away, it now strikes the roller, which results in the rollers being fed onto the tips of the cogs, several in a row until slippage occurs.
This is why modestly-worn "slipping" cogs often work fine with new chain after the hooked corners are modified with a 1mm bevel as shown on a single tooth of this cog: