Kinda bullshit, kinda not. It was always an urban legend until Batterygate a few years ago.
The nuance a lot of people missed there, was that older batteries would just completely shit the bed and shut the phone down if the phone pulled too much power from them, so Apple throttled the CPU down to prevent that power draw. On the one hand, they could/should have just alerted the users that it was time to get a new battery (as opposed to replacing the whole phone). One the other hand, if convincing you to buy a new phone was the goal, allowing your old phone to fail catastrophically probably would have been a better play.