Hi,
The only reason for a battery on these outboards is to power the electric starter motor and the tilt/trim if equipped.
If you don't have an electric starter, the enrichment system does use a bystarter "like" enrichment system but it is operated with a pull-push cable much the same as a "butterfly" type choke.
If your outboard is equipped with an electric starter as well as a pull starter, the by-starter is defaulted to FULL RICH (starting enrichment) when the engine is cold and not running. There's no "choke cable" and the enrichment occurs automatically.
As a result, the engine should pull start easily even with a dead or missing battery.
After the engine starts, a dedicated "enrichment coil" under the flywheel generates voltage (approximately 12V.) that will power the bystarter coil on the.#1 (top).carburetor to LEAN OUT the mixture so that the engine will run properly and not stall as it reaches normal operating temperature.
As long as the engine is running, the voltage is always generated by the enrichment coil under the flywheel and keeps the fuel/air mixture in the correct ratio for the warm engine to run correctly.
When the engine is shut off voltage is no longer generated by the under flywheel coil and the bystarter will cool and return to full enrichment after a short period of time.
I hope this is the information you wanted. If not, don't hesitate to say so. Happy to discuss the subject with you further if needed.
Welcome to the forum!