I'd say take on one problem at a time, throwing parts at it and guessing isn't going to solve anything. If you bought the ski recently and it's now running crappy even after a rebuilt engine and 'rebuilt' carbs you need to look at everything as suspect or not done properly..
First thing, and it was recommended to you by
@mikidymac in post #16, you need to rebuild the carbs to the letter with genuine oem kits before going any further. Adjust everything properly after reinstalling, set idle, cables, etc then if you're still having issues there's the next step. New fuel lines, fuel selector, and fuel strainer oring are a good practice. If you've done all that, you'd do a fuel bypass to narrow it down.
Then there's next steps if you're still having run issues, clip back the spark plug wires on each end to improve connectivity from the coil to the plugs. Are plugs properly gaped?
It wasn't having a starting problem before, so you need to back up and figure out why that's happening now after you pulled the carbs off, but again, I'd go through the carbs do a full rebuild with new oem kits. why? if you can't verify for sure..
You never did a test to rule out if the rectifier is bad or not, it was only an assumption someone else having similar symptoms it would fix it. The '12 volt low' is perfectly normal while cranking, even with a new fully charged battery, it'll clear up once the ski starts. The info gauge picks it up when below 12.4 volts, which it'll dip below that while cranking, not a big concern.