First, check your battery cables for oxidized and/or lose connections. If you have a voltmeter, your battery at rest with engine off should measure at least 12.5 volts.
Check all your fuses, there is usually one in the e-box where your ignition coils are, so check there too. Recharge your new battery only after disconnecting it from the ski, if the battery voltage is below 12.5 volts. Connecting your ski's MPEM to a power source greater than about 13.8 volts can damage the MPEM, many battery chargers can exceed this voltage. Therefore, disconnect the battery during charging and measure it before reconnecting to make sure the voltage is not too high.
Once the ski is back running, check the battery voltage at engine idle on the trailer, or 3000RPM in the water, it should be around 13.8~14.4 volts with the engine running. If yes, then the battery is receiving charging current.
If all this checks out and the battery voltage isn't above 14.4 volts, then there likely aren't any problems with the rectifier. A defective rectifier can in some instances cause the ski to run horribly, if the system voltage is too high or if the rectifier is passing AC instead of DC, but your description sounds more like a lose connection or blown fuse.
Never jump start your ski from another source such as a car, the system voltage can be too high. I would measure any external power source before connecting it to the ski, it should be less than 13.8 volts otherwise the ski's MPEM can be damaged.
If you find a blown fuse, and it blows again immediately when you replace it, don't install a larger fuse than is supposed to go there. If the fuse repeatedly blows, you likely have a problem, and this can be traced based on which fuse your are experiencing the problem with.
