I'm glad someone else has this problem because I had the same thing with mine. I stripped and rebuilt the carb twice, replaced the pump and checked all the fuel lines and filters. It got to the point that I would kill the battery or cook the starter before it would fire. Same thing, dry fuel bowl and stubborn fueling. I solved this by installing a electric pump in parallel to the mechanical one. I have it on a momentary switch so use it to prime the carb only. Once primed it fires first turn of the key. Benefits are if my mechanical pump ever fails I can very easily wire the electric one in order to get home. Not perfect but it works for me and it's similar to the old heat coil on a old diesel.
This is what I have considered doing it. I have been trying to figure out just how to plumb things in though. I am not sure if it is possible to put the electric pump in line with the mechanical pump, thereby using the electric pump to prime the mechanical, or if I should run a "T" in the supply line, and run a separate supply to each pump, then another "T" at the carb for each pump.
Some of my thoughts are: #1 If run in series, will the mechanical pump be able to draw fuel through the electric pump if the electric is not running.
#2 If run in parallel, would the mechanical pump force fuel back through the return line and then backwards through the electric pump if the electric is not running.
I know,

why is this guy going through all this convoluted plumbing?!! Why not just go electric only and solve all the problems at once?

. Because that would just be too easy! What else would I do with all my spare time?
But seriously, I kind of like the idea of a redundant fuel system given that we do go well off road, and often out of necessity, we go by ourselves.
Flex: I have the Motorcraft 2100 carb. I don't think the Weber 38 was used on the V8's.