Must say, even after rereading this thread. I never saw anywhere where either batteries old or new were properly load tested to verify good rebound.
I just think spending any time at all doing other than that is unwise. Go back and verify resting volts, volt drop when hitting the starter, then rebound volts after a few minutes rest. Those numbers tell where to go for faults and key to finding culprits.
I have indeed seen NEW batteries that do not meet specs. Though rare, a load test can prove the right in every case.
Once that's done, then you can confirm what the charging systems is doing. Get that spec.
THEN check the starter relay and starter.
Sometimes on an poorly serviced airhead you can find weakenesses as you go everywhere through the machine. It's a synergy of weaknesses adding up to poor and non function.
But if you assume a new battery is good then go on about troubleshooting for days and weeks, best get back to PROOF of VOLTS/AMPS in the battery. Has not yet been as far as I could find. A multimeter and 5 minutes is all that is needed to rule out the battery as a contributor in the equation.