How to add captions to a video on iPhone

Install Handcut from the App Store and open it. No account is needed.
Tap Add Video and pick a clip from Photos.
Open Auto Captions and tap Generate Captions. Handcut transcribes the speech on-device and times each word.
Tap any word to fix it, choose a Style, and drag the captions where you want them.
Tap Export Video. Keep Burn in captions on to put them in the video, or turn it off to get an SRT file instead. No watermark is added.

Free includes a 1-Day Free Pass (unlimited captioned exports for 24 hours from your first one); after that, Pro Lifetime is a one-time $9.99 unlock.