You are not unreasonable to request that they don't use your child's image in photos or videos.
What are they using the photos and videos for? Is it for sharing the children's day with parents and guardians? Promotion on social media, print media etc?
They should have safeguarding policies and training in place along with their risk assessments so that names and faces aren't shared publicly.
There is the additional concern in the current tech climate about data farming and AI which is increasing the levels of concern around the sharing of children's faces.
Children who are being fostered, adopted, or any other looked after children, will have very good reason for not wanting their face photographed or videoed to be shared.
It is very, very hard in the moment to try taking photos or video of a group of moving children if several aren't allowed to be photographed or videoed. Yes the emoji stickers can be used on faces, that's extra time and energy on top of everything else and it's much more time consuming and challenging if it's on video.
I can see you've had some very emotional responses from parents above who want to and like to see photos and videos of their own children taking part in school, holiday clubs, activities etc.
I think the suggestion that the setting has put forward briefly removing your child from a space or activity that is being photographed works is a reasonable compromise and your inflexibility is why your post has annoyed people.
We have changed our approach in school so any public facing images or social media doesn't include any faces. I had a similar approach in my last job which included promoting workshops and opportunities for people of all ages. Also on the blog I used to run and the connected social media accounts. Not using faces did impact negatively on how my content performed but there we are. I'd use images of children's hands in a tactile activity, or of moving feet in a dance class. It's enough to suggest the activity and the age of the participants but stopped there.
Photos are easier to ask people to step out of as it's a couple of seconds at most. How long are the videos and what are they being used for?