Since you're doing it by hand anyway you could add a set of valves to backflush valve A. Take water from your supply, for instance, and use a set of valves to isolate and inject water between valve A and the storage tank, then open A and B. You're reversing the flow to carry out the debris. You would have to have additional valves to do this when it was actually raining though, and I'm not sure how easy it is for you to add valves to isolate the different parts of your system. You might be able to do it with minimal digging. This is basically what you do to backflush an irrigation pump that gets water from a pond or something where it can pick up crud, or it's one way anyway.
The drawback of the downspout first flush systems is the difficulty in diverting enough water to do the job if you have a sizeable roof. You're lucky to divert 10 gallons or so if you just use one 4" diverter pipe, assuming a one story roof. I've heard you should try to divert 1 mm of rainfall, so I should be diverting 30 gallons or so from my 1200 sq ft. roof. It's better than nothing, though. I think that's right, it's been a while since I figured it out. I considered adding a small storage tank in parallel to the pipe, that could work if it is done right. I guess you could have 4 diverters, seems silly though.
I'm not a expert or anything but the fine dust ending up in the bottom of your tank is probably just tree pollen and crap like that, not horrible. Every tank has biosludge in the bottom eventually and in fact I think it helps keep the water clean. The trick is not to disturb it.