Not only do most teams have closers with very defined roles, but it's probably also true that in this day and age of monitoring pitch counts, more teams have a better overall bullpen than teams from >35 years ago.
I absolutely fondly recall the days of starters having more complete games like you, but the best explanation I've heard from modern-era managers is -- even forgetting about pitch counts -- they have to weigh whether or not they think the 107th pitch from their starter in the 8th inning is going to be better than the 1st pitch from a quality, rested reliever.