here I soldered the three lines together and heat shrunk
Perfect.
How that all works is that the 2 yellow wires are each end of a few coils under the flywheel. The flywheel has magnets in it, and as they pass the coils it induces an AC current. This current is converted to DC (by use of a rectifier) and feed to the battery. The early outboards only had an output of about 6 to 9 amps, but the later ones, including yours, is about 20 amps (for each set of coils, you have 2). With the higher current the battery would be over-charged very quickly. Even with the lower current is was not unusual to see batteries at 17 or more volts. Normal automotive batteries couldn't cope and often boiled dry and could explode, so that's why we only use proper marine batteries which are designed to cope with being overcharged (and the fact that the plates are properly supported to cope with the 'rough ride')...
Anyway, the higher current output called for voltage regulators to be used. In cars, the voltage is controlled by the current fed to the rotor in the alternator (that's the bit that is the equivalent of the flywheel magents), but as the field strength of the magnets in the outboard can't be varied, and so the output can't be reduced as the battery approaches full charge, the excess current is dumped off as heat. Which is why those regulators get very hot during operations. Some models even have the regulator/s bolted directly to the engine block so the heat is pulled off into the cooling water. The regulated output is on the red wire, which feeds directly to the battery.
For the tacho, the early systems just was one of the output wires as the feed for the tacho, and the tachos has an input marked '12 pole', which is the number of pulses per revolution from the AC side of the alternator output. For some reason the later engines included a tacho output from the regulator (the grey wire) which came from some fancy bit of circuitry in the regulator. When they inevitably fail, we just go back to the old tried and tested tacho feed, the alternator AC output, one of the yellow wires.
Cheers,
Chris........
... Usually just toss them in the trash and buy a new one. Not expensive. What is the gauge series? (Commodore, Admiral, QSI, Flagship, etc) Post a photo if you're not sure.