That would seem to be OK for a carb engine (where only a minimal head is needed to feed the carb which has its own float valve) but how would it work for a typical fuel servo for say an IO320 which needs a lot more fuel pressure?
Gravity never fails but you don’t have an inverted fuel system this way so can’t fly upside down (for long) unless you keep +ve G 
Also 20-30 mins is cutting it close if you are over the Alps.
Peter wrote:
Also 20-30 mins is cutting it close if you are over the Alps.
Yes, but it’s a lot better than 2 seconds.
I am not sure the probabilities are comparable.
Normally you have the engine driven fuel pump. If that fails (which is very very rare) you switch on the electric one. To get the failure of the former, plus a failure of the latter, would be amazingly bad luck (unless the latter was duff before the flight, which would be truly dumb, and it would be very hard to start the engine).
With the header tank system, you don’t have the engine driven fuel pump, but when the tank runs low, you turn on the electric one and find it has failed, what do you do? You have only a single point of failure, plus whatever fuel is in the header tank when the failure of the electric one is discovered. One would really need two pumps to refill the header tank, to get an equivalent reliability. Or, if you want to make a trade, you could top off the header tank when it is say 90% full, and then a failure of the single pump would still leave you with 90% of 30 mins (or whatever).
What I still don’t get is the fuel injection scenario. How do you feed the RSA fuel servo? There must be another pump in there somewhere, with a spring loaded bypass so as to deliver a constant pressure. OR a return to the wing tanks. OR most likely the IO320 has the normal engine driven pump, like most Lyco engines have.
Googling suggests the fuel servo needs 20psi which is way more than the head from a header tank.
Looking at the document, the default setup seems to be engine driven pump + electric backup, taken from the header tank. Then transfer pump(s) from the wing tanks to the header tank.
The Australian guy I linked to made his own setup with a valve, so the engine driven pump could take fluel from either the header or each wing tank. In addition he had transfer pumps. I don’t know what the common setup is amongst Lancair owners/builders, but if I were to build one, I would think in directions of simply removing the header tank altogether. Having a header tank in the cockpit if there is no good reason for it, seems like a strange thing to do. If you need the extra range, I would simply gravity feed the header into the wing tanks instead. But then, maybe there are some CoG issues here also?
I’d guess Lance N. and company borrowed the Dragonfly fuel delivery design (single pump to header tank, then gravity feed) without considering injected engine issues. The first Lancair had an O-200 engine and the next an O-235, neither engine is available with injection.
