Tiny fuel line leak? After running, fuel in the rail & upper line boils away through the leak. It takes 15-30 seconds to refill and pressurize on starting and I think that for safety reasons the pump doesn't run continuously unless you're cranking.
Been there, done that. On my car the fuel filter is on inside of frame in engine compartment, the bottom connection is invisible; it's a flare nut. When I took out the stuff needed to reach down there and feel the connection it was wet -- but not leaking fast enough to drip on the ground so the usual inspection would find it. Fixed the problem and -- Always varoom! on the second revolution.
After starting and running a minute or so, inspect all fuel line joints -- there won't be more than about four-five up to the rail plus another one for the recirc line. 'Bone dry' is the only correct answer.
'Priming' should have exposed this issue but you need to keep that pump running for quite a while. On my car there's a convenient connector where you can hook +12V to run the pump regardless of the rest of the system but if you don't have that, who knows what it takes other than a bunch of cranking.
Or, of course, I could just be wrong.
If it is a leak there is a fix regardless of the flavor. You can get copper gaskets for flare fittings.