I think that method was used much more widely than just the Glasgow Cup, and until well into the 60s at least. It was abandoned as being too prone to lucky breaks. For example, a team could dominate for 95% of the game and have 20 shots on target or that hit the woodwork and score once, but without getting a single corner, while the other team might have attacked once, got a single corner from which they scrambled a goal, and then getting awarded the game because only they got a corner.
At least penalty shoot-outs require some skill, by both the taker and the keeper. Luck too, of course.