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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.

This is incorrect. The setting does not apply to Safari. It's for App Store, for example.

Now tested - you’re right

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

My guess is no.