Very first place I'd start, before any other guesses or actions, would be at the battery terminal connections. Remove each one, clean to shiny metal on connector and terminal, reconnect, move to next one.
I got an emergency call one year from my Mother. She couldn't get the generator started and lights went out. She wanted me to come and replace her batteries.
I stopped in, took one look at the batteries, then proceeded to do that I said above.
About 30 minutes later the generator fired right up. FWIW, the battery terminals didn't look all that bad as her husband (who was at the time in the hospital getting a cardioverter installed) had been doing the old "Baking Soda Bath" a couple times a year. Inside of connectors were so corroded I had to use a pocket knife to peel the corrosion off both connector and battery terminal.
It's not uncommon to read full voltage but once high current of a starter motor hits, the connection breaks.
Clean terminals will provide quicker charge times and more reliability. Start there first. They'll need to be cleaned anyway.
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