Weller EC1301a vs EC1301b (or schematic for EC3001 station)
Incidentally, I came across this same soldering iron a year back at a garage sale for a steal, but it has the exact same issue you describe: powers on just fine, indicator light stays on, but tip stays cool. Can't find much by googling myself, but it seems other Weller models from the era with the same problem have been fixed by simply replacing the tips (not the pen). I'm waiting for my replacement tips to arrive, but thought I'd share in case that helps.

Sources I've found helpful:

https://forum.sparkfun.com/viewtopic.php?f=14&amp;t=20773&amp;start=0
Older Weller soldering iron won&#39;t heat?

Also, this site has a wealth of old Weller parts manuals:
http://arcarc.xmission.com/PDF_Misc/Weller%20Parts%20Catalog.pdf

Hope that helps!