Ah the wonderous Magnastat, I still use my TCP every day and it's still more effective than a lot of the modern electronic ceramic element ones.
The tips themselves aren't magnetic, they are ferrous alloys. The magnet is inside the therno-switch assembly down the center of the element. The slug on the back of the tip attracts the magnet until it reaches it curie point temperature, when it stops attracting the magnet.
If the magnet isn't being attracted (you should hear a click as you insert the tip) then either the thermo-switch has been pushed too far back in the barrel and stuck, or the plunger in switch assembly itself has stuck. If you hear a click when you insert a bit, then it's the switch contacts or some other wiring fault.
If something has stuck (and the iron hasn't been used for a while) then try tapping the iron with the tip removed end downwards onto a piece of wood. That might be enough to loosen thing up again.
As far as I'm aware, there's no age related deterioration of the slug on the tip, it's curie temperature is set by alloy composition only, so they should be fine, regardless of whether the iron works.
BTW: I fitted my TCP with a 5 core lead and use the termo switch to control a SSR in the base unit. That removes the contact loading from what is the most expensive component in the iron (elements, handles etc are pretty to very cheap).