that chainring would have to be extremely worn to allow chain over-ride slippage.

the most likely thing is a Bad Freewheel ratchet that's not engaging. I see this frequently, and especially on low priced bikes or ones that have been left outside for extended periods of time in the rain.....
next most likely is a bent Derailleur hanger caused by a crash or dropping the bike to the right side...

Outright Chain over-ride is the least likely cause of the slippage, but can be more likely caused by a Stiff or Twisted Chain.

and yes, a "new chain" can cause over-ride... if the Wrong Chain was installed, or the gears are super-worn out, and look like shark fins...that condition usually only happens to a couple of the middle gears in the rear, and is very rare.

other possible causes of over-ride are bent teeth on the chainrings (front gears), a Bent gear or two... and extreme rusting of the components.

Check for uneven engagement or slow engagement of the rear gears' ratchet mechanism (the "Freewheel") first... i've seen more than a few of those literally break only a few weeks into the life of the low priced bikes being dumped on consumers.