Yes a dead/dying battery will generally upset an electronic ignition, Boyers do require a minimum voltage to start with.
Also the alternator rectified output is not a "clean" DC supply and tends to come in pulses which the electronics don't like.
The battery tends to stabalise the ripples & clean up the supply. While the origional contact breakers may well run on alternator output (probably not well) I think it would be a backward step to revert to them in terms of accuracy & reliability.
I would reccomend as the first step to check that the battery is actually charging. Put a voltmeter accross the battery, check for 12v plus then start the engine & look for an increase in voltage that rises with engine RPM, no increse = not charging.
One option you could consider is running a battery less system. I work overseas alot & got fed up of replacing batteries after long lay ups. Both my T140 & Tr7 are wired with a "Mitty Max" battery eliminator, one runs a boyer & the other a Lucas Rita ignition. Both normally start 2nd kick & have had no issues for several years now. If you do this I would reccomend changing the lights to LEDs to reduce power requirements & remember to wire it NEGATIVE earth as being diodes LEDs only work in one direction.
The Mitty Max is solid state electronic unit with a built in capacitor to stabalise the alternator output. It also replaces the old rectifier & zener diode regulator.
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