Does anyone know of current startups in this space? The last high-profile development was the Segway, and that clearly didn't catch on for its intended purpose.
That said, it sounds like you wouldn't have to worry about line changes and transfers with PRTs. Which would be nice.
I think the only way you can hope to achieve this is through zoning and long-term alteration of how cities are built. Sprawly cities like Houston et al. can't possibly hope to build out any right-of-way based system such that everyone is within a mile of a station...thus you build the center and let density rise. Combine this with a suitable number of edge-sitting mega stations that outlying folks drive (or, even better: ride a bus) a shorter distance to (relative to a full commute) and you've solved a large portion of the problem, anyway.
Which "doorstep-to-station" issue? There are two ends to most trips. And no, work isn't always close to a transit stop.
And, there are trips that have three ends, the "middle end" being the one that you make on the way home.
And, I wouldn't use the phrase "fairly competent" to describe a system that takes twice as long to get from point A to point B as many systems outside the "great cities" do.
If PRTs like this can be deployed more cheaply and widely than subways (one of their main goals), that means you can bring that down even more - perhaps 200-250m. There are very few people that cannot be served by system being a 3 minute walk away from them.