Remember that until the 1980's teams used a four man starting rotation and in the 60's and 70's generally only carried maybe 10-11 pitchers on the 25 man roster instead of 13. Pitchers finished what they started much more often so there wasn't the need for all the spot or specialty relievers. The top pitchers threw more than 300 innings, sometimes as many as 340 or so.
I still do not feel there were as many arm injuries in the 1960's as there now but I can't numerically prove it. Today, many pitchers are like cannon fodder with the massive numbers of Tommy John surgeries.