12.6 Volts should start it okay, but the problem smacks of ultra low voltage.
It could be getting stopped by a high resistance connection(s) somewhere? This would probably have to be at the battery terminals to affect the relays and the starter.
I'm thinking it could be that the battery has completely run out of "cranking amps"?
To test this, put a meter on the terminals and watch what happens to the voltage when you thumb the starter. If it's lost it's cranking amps the voltage will drop dramatically at this point and probably slowly recover when the button is released.
Only cure for that is a new battery, but have someone put a professional meter on to confirm. (Maybe at the battery store or a garage?)
If that cures it and you get her running again, I would check the charging voltage. The burnt relay smells of over charging.
Just put the meter across the battery terminals when it's running and you should see about 14 volts. If you see 15 volts or over at say 2000 rpm (or any revs!) then I'm sorry but your reg/rec is fried and needs to be replaced.
Make sure to use very low resistance connections on the 3 yellow cables from alternator to Reg/Rec, or hard wire them. I use Gold connectors on all my bikes and have trouble free charging.
Check the yellow cables for heat stress, soft, charred or melted insulation and replace as necessary.