is there a quicker way to add screen record on iphone?

If the button's there but nothing records, or it stops right away, look at Screen Time. Settings > Screen Time > Content & Privacy Restrictions > Content Restrictions has a Screen Recording option that can be set to not allowed. Kids' phones especially. Recordings save to Photos, by the way.

Put the button in Control Center so it's one swipe away. On iOS 18, open Control Center, tap the + at the top left, then Add a Control and search Screen Recording. Older versions it's under Settings > Control Center. Long press the button if you want your mic recorded too, it's off by default and that's what gets most people.