1st, in the electrical subforum here, post #13 (?) has a very clear wiring diagram. Download it ! It's a must have for dong ANY 'electrical'...
2nd, you should say which 'BX', as there's usually variations between them.....
3rd, to start a BX23S , you need PTO lever 'off 'disengaged' AND treadle pedal in 'neutral'. Turn key to 'start', engine spins over. That's it, don't have to be in the seat.
ideas...
have a look at the PTO lever, kinda easy to bump it,while getting battery in or out , and you wouldn't know you moved it.
'rock' the treadle pedal while key in start, see if starter spins.There should be a 'hole' in the mat to access the grease Zerk..,well on my BX23S there is, easy to forget..
verify the small blk/wht wire IS connected to spade lug on starter ! Mine 'magically' fell off at abour 5 hrs on the meter, Reconnect,tyrap, 1400 hrs since then...perfect.
Please try stuff, report back