Is there not a factory-original rectifier already providing DC output from that same alternator? Why not just move the power supply for the headlight switch to the existing DC bus?
On a lot of small motorbikes, especially those without electric starters, the electrical system is feeble. If there is a DC system, it is likely to be fed with a half wave rectifier, and it will only power devices that are used occasionally, like brake light and turn signals. There is often no regulator, and the battery, which as Nigel said is far too small, is just big enough to support the brake light and turn signals for a few minutes.
The ignition system is often a magneto type, where a coil generates enough energy for a spark in a single turn of the engine, so no connection to the DC system is needed for that either.
The same is true for some lawn mowers.
https://www.lawnmowerforum.com/threads/riding-mower-lights.8581/
It might be possible to completely redesign the electrical system to generate much more current, have a full wave rectifier and a regulator, as well as a larger battery. However it would be a lot of work, and who ever did it would have to be careful to make sure that the battery charging was adequate in all conditions. The scheme being discussed is much simpler, and all the additional circuitry will probably fit behind the headlight that is being changed. As long as it works at tick-over and at full revs, it will be fine and it won't matter what state the battery is in.
I've driven cars with dynamos where there simply isn't enough electricity being generated to allow extended driving in town with the headlights on. Alternators changed that by working with the engine at tick-over. However, well after that, motorbikes were still being made that wouldn't keep the battery charged when driving slowly and using the brakes and indicators a lot.
As for the correct main-dip pattern, when I had a motorbike (1983 Honda CG125), changing from the standard headlight to a Cibié one meant that dipped beam had a proper shape to it, not just the hollow circle of an out-of-focus lamp. It wasn't perfect, as the lamp was designed for driving on the right, and the bulb had to be rotated a bit for driving on the left, and the cut-off lines were a bit curved, but it was far, far better than the standard one.
I realise that motorbike lights, even the AC driven ones, have probably improved a lot in 40 years, but an LED headlight will probably have a very good beam pattern.