There are other reasons they want to disband the post office beyond just its impact on voting. If they replace it with private entities it becomes easier to surveil and censor the mail, for instance.
The USPS takes an image of the front of the envelope for every first class letter sent in the USA. It's a safe bet that they're doing OCR of the recipient/sender addresses and store it in a database. USPS Informed Delivery is just a customer-facing result of that long standing program.
The USPS includes legal protections against your mail being opened, and a mandate to serve everyone. Whereas private companies can be fully employed for government surveillance (third party doctrine), sell bulk records to commercial surveillance companies (equifax, google, etc), as well as refusing service to whomever they like (ala MC/Visa).
I agree this isn't the overriding concern of those trying to destroy the USPS, but it's surely a nice bonus.
There was this incident a while back. Some psychopath sent letters threatening to kidnap young females, rape them, and kill them. He talked about how he had rigged out a van to facilitate this.
His letters got turned over to the FBI, since they would up in different states. Turns out that each stamp has microprinting on it that identifies it. Given the stamp, they were able to identify which post office it was sold at. They guessed at approximately when it was sold, and then from the security cameras they identified the letter writer.
So... explain how private entities are going to do more surveillance?
Private entities will just have terms of service which say they're entitled to open and inspect your mail for any reason they see fit. mindslight's comment above does a good job of touching on some other ways it will enable more surveillance.
But yes, the postal service has already been turned to this purpose in many ways and the days of "gentlemen don't read each other's mail" are largely over.
