I wonder how much of the "managing processes" part could be delegated to software tools? How many of these processes could be automated, or "managed" by some sort of workflow software. Would it be possible with the right software product to essentially distribute the role of manager across the team/company?

The problem is that a lot of processes still leave a lot to interpretation, or many scenarios have grey areas on how to apply a process. The author mentions it a few sections later, you don't want the processes to have a ton of overhead, rather make it clear the expectations on how to interpret the situation or know they should come to the manager to provide direction.

Yes, but some processes ARE well defined, and you want to ensure the steps of it are all met correctly each time. This could be similar to the checklists airplane pilots use, or surgeons as extreme examples.