Depends on the battery, and the PO as well. Could be his version of "new" is 6 months to a year old thinking like the life of a car battery, but if your battery is a flooded lead-acid version like the bike came new with 50 years ago then they don't always last very long depending on how it was treated. With these charging systems and a flooded lead-acid battery, the battery has to be checked at least once a month and topped off with distilled water or it will slowly boil off electrolyte and if it's run low on water/electrolyte for very long, it's shot. If it's a gel or AGM battery then the lifespan is far longer. As to the low battery and the bike running on it if kickstarted, yes it will if the charging system is working AND you go ride it right away because again, these charging systems don't recharge the battery at idle. So, if you got it running with the kickstarter but sat there with it idling for very long at all, the battery only gets lower.
If the electric start makes noises like shaking a can full of nuts and bolts, then the starter clutch needs to be rebuilt. Generally speaking, these elecvtric start motors last a long time and the starter clutches are the first wearout item.