Starters draw lots of current, hundreds of amps with the initial surge. While the battery might read good with a DVM when not starting, once you put the load of the starter on a dying battery it's deficiencies come to life real quick.

When the battery went out in my MIL's car it read good with the meter connected to the battery, told her to turn on the headlights and the voltage dropped way low instantly. It was bad.

Fully charge the battery then take it to a parts store for a load test. That will tell you what's up.