Being 1 link too loose is enough for low tension to be a problem.

Bicycles that are designed for single speed usually either have horizontal dropouts or track ends, allowing the wheel to be positioned so that the chain is sufficiently tight, without using a tensioner. This is why vintage road bikes are popular for 1x conversions; they just about always have horizontal rear dropouts.

If you don't have horizontal dropouts, then you either need a tensioner, or some other mechanism like an eccentric hub. Or, if you're really desperate, there are some kludgy solutions like fudging with the sizes of your cog and chainring to adjust chain tension.

This page is a good reference.