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Settings, Accessibility, Motion, switch off Auto-Play preview videos is supposed to do the trick for System Apps (including messages, safari etc).

Untested since I run my phone via Wireguard to my home network and block everything there.

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I’ve always wondered about this, does it stop downloading the video when it stops playing?

My guess is no.

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This is incorrect. The setting does not apply to Safari. It's for App Store, for example.
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Now tested - you’re right
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StopTheMadness will do this pretty well for $15.
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