I suspect the primary issue is that there are simply way less people willing to work as letter carriers. The pay has not scaled and you're working long hours, Monday through Saturday. Not exactly a dream job.
This describes my federal job. I wonder how many other federal jobs are like this. My coworkers used to put up with the stress/schedule/etc, but now the pay does not outweigh the cons. I foresee the industry I'm working in, becoming a shitshow; I wonder what other industries will follow suit.
A family member of mine works for the post office in their city. They don't have trouble hiring for any positions. There is a long waiting list that takes months to get through to be given a chance at the bottom of the totem pole. The various postal unions give the jobs good benefits, pay, hours, etc. Carriers specifically are highly sought-after positions that can set you up for life.
It's not a matter of not being able to find people. it's not willing to hire more people. I would be very surprised if the post office in your town is trying to hire people, but nobody is willing to work. It's a great job, and people line up to work there.