Apple TV audio sync not playing audio from wireless speaker

No worries. Now that you've clarified I remember doing this testing and actually that is a normal part of the procedure. The tone will play through the TV while the phone is being held close to it, it is also best to have the TV's volume set at an appropriate level and have a quiet room. I have posted the support article so you can follow the steps.

Set up Wireless Audio Sync with Apple TV - Apple Support

This is a user forum, you are not addressing Apple here.

When you say wireless speaker are you referring to Bluetooth or Airplay? If bluetooth, have you ensured it is properly paired. If airplay, have gone into control centre on your IOS device (or airplay icon via Computer) to initiate, and have you ensured that it hasn't cut out?

It is using your phone to measure and adjust latency. If it says that the process is complete then it was successful, you would only need to redo it if says it failed or if you replace a component.

If it was successful and you still experience latency then there may be another factor that cannot be compensated for, such as further interference or a problem with the speaker.

Airplay is superior to bluetooth in general and I can attest that Sonos makes quality components. For regular use I would recommend their Beam instead as it is better to have a wired soundbar for consistent output that is tuned for both music and TV.

Understandable, but keep in mind wireless still relies on a consistent and reliable connection and when using it for video on a regular basis it can be more prone to have issues with sync. This feature aims to fix that, but it cannot account for everything nor can one expect miracles. These are also all mono speakers so while it still works great for music, and will sound better than in built speakers for standard viewing it won't be as great for movies. A soundbar will re-route all audio, whereas wireless can only work through Apple TV or other specific device that supports those protocols. There is also the option to mount the Beam if one is limited on space.

With that in mind, I would still recommend the One over any BT speaker, they also have great CSR's and if you buy directly through them they have a great return policy should it not work out.

As I said that is the normal part of the process, it is using your phone to measure and adjust the latency. However, if your intent is to use it as a permanent speaker for Tv then that is going against the purpose so you may still run into issues. If you still experience the latency there may still be things to troubleshoot, but would need to cross that bridge when we come to it