Kent Shultz
Maybe its time to step back and consider things. If I recall correctly, neigbor's 582 won't start, brings it to you; you deduce fuel pump and/or carb issues. You've rebuilt fuel pump and carb, engine still won't start.
Three things necessary for combustion: fire, gas, compression.
Have we positively eliminated spark as an issue?
Have we verified compression (both cylinders)?
I have some sympathy for your situation. It took me a year to get the same engine running correctly. I had issues on all three systems: mechanical, electrical, and fuel.
Tell-tale sign of compression issue on startup: spins too fast, no "grunt" to get over compression stroke.
Tell-tale sign of fuel issue on startup: motor will start with ether (starting fluid) but won't stay running (because its not receiving fuel delivery).
Tell-tale sign of ignition issue on startup: (1) elimination of the other two issues, (2) no fire under test conditions, i.e., spark tester, or screw-driver 1/4" away from block, or Kohler rig with two spark plugs minus the electrodes (see their manuals).
A final possibility is that the timing is off. This can happen when the crank pin is sheared or partially sheared. Under this condition, the motor passes all the tests above (except running on ether), but still won't start.
The only way I know of to investigate the timing is to remove the flywheel and examine the key. You might need some help holding the flywheel while getting the nut off, it is torqued to 140 ft-lb. (Briggs & Stratton sells a special wrench for the purpose, or you can use a large, sturdy strap wrench, or a screw driver on the blower housing at the breather hole --but you may damage the blower cover.)
Then you still will need a fairly strong puller to remove it. (I used a harmonic balancer puller with bolts from a steering wheel puller.)
Good luck.