On the subject of tightening a loose tenon, I wanted to share my tools. If heating won’t swell it and beeswax won’t do the trick, you want to expand as much of the length of the tenon, rather than just heating and pressing it to flatten the tip only. I have several small dowels bought from the hardware store. The pic shows my arsenal along with a toothpick for size comparison.
I cut a small section and then sand it to gradually increase the diameter, much less sharp than a pencil. The ultimate goal is to heat the tenon and insert the dowel to swell the tenon along its entire length, not just the tip (which a sharpened pencil would do). I put the appropriate size dowel in the cold tenon and sometimes make a mark to see how deep it goes after I heat it up. Of course you want to barely press it, rather repeat several times to get the correct fit.
Do not try this with a matchstick that you’ve sharpened. It will break off inside the tenon and be difficult to remove.