Although the relationship between the two sprockets remains the same, I think the chain moves along the teeth in the same way as the timing gear teeth do, only ending up where they started after several turns. A dab o' paint would prove this, must admit something I have never bothered about.
Even new chains and sprockets are adjusted at their tightest point, and the old adage of all new or nothing applies, worn sprockets will wear a new chain, and an old worn chain will wear the new sprockets.
Make sure the new master link is an imperial size, not some parts man idea of a metric equivalent.
Sadly comparing the image of the rear sprocket teeth with those of new parts on the web, those are definitely past their best, showing a sharp point and more wear to the load side of the teeth. This will go a long way to explaining the slack-tight you have. Having said that, there is still some life left, provided the chain is adjusted at the tightest point. We've all seen worse....An estimate of how bad chain and sprocket wear is can be to simply to lift the chain away from the sprocket, while pulling on the chain run. A good chain and sprocket will be nigh impossible to move, a badly worn example will lift almost to the top of the teeth.
Swarfy.