apple tv 3 after reset the setting does not work airplay

Does the Apple TV work for other streaming services (non-AirPlay)?

Does AirPlay/Screen Mirroring work for other content (different app, or even the home screen)?

Verify if you have the dreaded “Activation failed”-error in Settings﹥General﹥About﹥.

Use AirPlay to stream video or mirror the screen of your iPhone, iPad, or iPod touch - Apple Support

If screen mirroring or streaming isn't working on your AirPlay-compatible device - Apple Support

activation - failed

Solve that first, as I think it prevents AirPlay as well.

there is no airplay icon on the phone

Some iOS apps have in-app AirPlay/casting controls. So that depends on which app you use.

For screen mirroring, iOS Control Center has a button for that.

Wirelessly stream video, photos, and audio from your iPhone to a Mac - Apple Support

Activation is about signing in with your Apple ID account, so that the device is registered to a user in Apple’s administration. There are a few other threads about failed activation on this forum, that suggest unofficial remedies.

I would suggest to contact an expert from the Apple Support staff for further assistance. Contact Apple for support and service (phone numbers) or start your support request online (chat/​email/​callback).