A person CAN put in serious road mileage on the H-Street but I don't know if you can. It takes a while to work up to and I haven't reached the point at which I can do it. I run in the H streets for medium/short runs on the road and any distance on dirt.
If you've been running in DS trainers then you're probably not ready for them full time but you could start transitioning. I would look at a flat with a little more to it first. A shoe losing it's cushioning does not become a minimalist shoe. Rather, a minimalist shoe can lose its cushioning because it is not dependent on it for function. You follow? The problem with wearing out trainers is that so many of them have dual density posts. This means that the lateral edge of the midsole wears down faster than the supportive, medial edge. This can make the shoes very dangerous from an injury standpoint. A truly neutral shoe should wear longer because it will wear more evenly.
Last word: if you wear the DS, a logical next step might be the Tiger Paw. That's what used as my first training flat.