Continuity in comics only really became a bigger deal in the late 1950s when all the various elements established in the Most Weisenger edited Superman comics solidified into a pretty complex continuity (even though they often contradicted other DC comics and older Superman comics) . The Julie Shwartz edited comics of The Flash, Green Lantern, The Atom, Hawkman, and Justice League of America also had their own tight continuity.
However, it wasn't until Stan Lee, Jack Kirby & Steve Ditko took that concept and really blew the lid off of the idea of a tight inter-connected continuity at Marvel in the early 1960s that it became a bigger priority for DC to follow suit because, up until that point, the market was children who would cycle through every few years so it wouldn't really matter if newer stories contradicted older ones.
The introduction of the Multiverse in the early 1960s allowed DC to continue to play fast and loose with their continuity for decades because anything that didn't fit could be explained away as an alternate Earth. However, with Marvel's more connected continuity resonating more with readers, the original Crisis was DC's path to follow suit.
Dick Giordano was less concerned with continuity than Mike Carlin, who applied a much heavier hand in keeping everything straight, but the problem he had was that a lot of the decisions made for the Post-Crisis continuity were bad ones and his insistence on sticking to those dumb ideas were largely a waste of time because creators kept trying to reverse them.
Dan Didio has had a much more chaotic idea of how DC continuity should work, as evidenced by his repeated desire to reboot the DCU until he finally got what he wanted with the New 52, only to watch it blow up in his face.