In my opinion based on my experience: it depends.
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With enough tension the chain doesn't slip on that cog.
The chain is so elongated that a chain checker doesn't fall into the gap between the rollers anymore.
But don't try this at home, this is just an experiment.
The old 6speed drive trains surely don't wear down as fast as more modern ones, because the teeth on the cogs are higher, the cogs are thicker, the chains are wider and therefore there is less pressure at the critical points in the whole drive train. Well maintained that stuff lasts for ages.
The old cogset differ from manufacturer to manufacturer and many of them are no more, eg. Regina or Maillard. Please be aware that there could be different thread types for the hubs - italian and BSC were common on track hubs. On road hubs it can be similar.
My approach would be to invest in a new chain and try to use it on that cogset. If everything runs smoothly, then I'd purchase at least another chain and change the chains regularly, because the cogs and the chainrings adapt to the elongated chain - so the chain is the culprit for "deformed" teeth on cogs and chainrings.
In my opinion changing the chains once or twice a year would be sufficient for that amount.
A chain for 7/8speed will work on 6speed cogsets - regarding to the width of the cogs.
Hope that helps,
E.