Could be a lot of things, but don't overlook that you don't really know the quality of your replacement battery. It would be a shame to screw around with everything else and end up back at the battery.
A load test on the battery tells you reserve cranking capacity, a static voltage check isn't sufficient to determine that.
Beyond the battery, a bad connection, ground or otherwise, bad stator coil, or bad rectifier/voltage regulator is what you might end up dealing with. A weak magnet which spins inside the stator is an unlikely last resort.
I suspect your battery, those bikes use a lot of current to be electric started, and don't have high amperage batteries to
start with.
A dead battery can still have enough power to illuminate a couple of light bulbs.
But it won't have enough power to spin the starter motor.
Little bulbs can be lit with a AAA battery.
After charging a dead battery, it will have enough strength to run the electrics.
But without the charging system charging the battery, the battery is going to drain down to zero volts (dead).
Check the charging system.
They go bad on all bikes.
I had an old Honda that did the same thing. I bought a new battery but that only fixed the problem temporarily. I went to the library and copied the diagnostic section of the manual. I went home with my multi-meter and within an hour I found a dead voltage regulator. A good manual is a great investment, and HD sets the standard for them.
Timbo has the correct next step - the charging voltage should be > battery voltage and < 15 V. Check idle and about 3000rpms if it drops off at the higher RPM to battery voltage you have a problem - if it never charges above battery voltage you have a problem, if it only charges above battery voltage at idle you have a problem.
The spare battery you got may have a dead cell, so the voltage is too low for the starter to engage properly.
Get the battery tested or just buy a new one.
Check alternator output as it may not be charging the battery.
Your stator is bad .