Andrew R Stewart 

Worn chains and cog skip under loads are completely different issues from the tight/loose cycles of chain tension in a non spring loaded tensioner system (single speed or fixed gears). Also ring lateral "trueness" has nearly nothing to do with chain tension. Lastly was the old ring an even tooth count one? If so then no wonder the teeth wear pattern is alternating, the chain has inner then outer plates which contact the teeth. An even toothed ring will have the inner plates only touch every other tooth. Had the ring been an odd tooth count I suspect the wear would have been more even (bad pun) Andy