Can't Tighten Handlebars
I'd guess that there are two possible causes.


Old age.  Your wedge as pictured shows some clear flat spots on the ridges that are supposed to bite into the inside of the steerer tube (the receiver that forms the top of the forks)   So its very likely the steerer has equivalent wear on the inside, and you're simply seeing a lack of grip.

First fix would be to file the wedge so its got more obvious ridges in it.  Use a triangular file, or the edge of a quarter-round file for metal.  Do be conscious of metal shavings and avoid getting them near bearings.


Another repair plan would be to buy a new wedge - this probably means a new/replacement quill stem.  If there's a bike coop near you they might be able to help.


The other cause is damage or wear inside the steerer tube.  This is impossible to fix, you'd have to replace the front fork completely.  To check, you need to undo the threadded headset and drop the fork out the bottom of the bike.  

You're looking for bends or swellings or even holes in the outside of the steerer tube, or significant wear marks inside.


Dropping out the fork also allows you to service/replace headset bearings too and clean out the bearing races, which may be due given its age.