Why did Axial Capacitors Fall Out of Use in the Industry?
PCB area. 

Axials pre-dated PCBs, their construction was ideal for wiring to tag strips and valve bases, and they were adopted for PCBs because that's what was available. 

Example of tag strip construction below.

(There WERE radial caps in the valve days : they were generally designed for chassis mounting via a ring clamp, and had tags rather than wire leads. The round object bottom centre is the base of one such capacitor)



It's actually quite surprising they lasted as long as they did alongside radials, into the 1980s.

Radials use much less PCB space, and standing axials on end is a poor compromise, with a long exposed lead (or the added assembly step of sleeving it) as well as being much less robust.