1963 Valiant 'vert door latch latch (?) not working

I completely forgot one other possible issue. If your lower door hinge is worn and the door is sagging as a result, then the latch and striker may not line up. So even if your striker is in the position it's always been in, it may not be in a workable position now.

Easy enough to check the hinge: with the door open, grab the door handle and see if you can lift the door any significant amount. If you can lift, then drop, then lift, then drop, your hinge is worn. If so (and in my experience it's always the lower hinge), best to repair or replace it. However you may be able to get a temporary work around by adjusting the hinge and/or the striker.