I put a new battery in my 2011 Macbook Air (turned out it was the power supply that was dead, but that said, the battery had been flashing up a health warning for a year and ran to about fifteen minutes). It was a third-party but I paid modest premium for a known brand* (as an ancient computer I don't especially need, I wasn't bothered about a proper Apple battery though if it was a mission-critical computer I'd go for genuine).
Works fine, gives it about 2.5 hours of charge time (which I think is all it ever got, things have come along since 2011).
I think a some laptops require a battery (even a sickly one) to be present before they'll boot even on a/c.
*not sure I'd trust any 'genuine' battery from eBay.