ChatGPT Search not working and web icon missing

Based on Introducing ChatGPT search:

All ChatGPT Plus and Team users, as well as SearchGPT waitlist users, will have access today. Enterprise and Edu users will get access in the next few weeks. We’ll roll out to all Free users over the coming months.

I am not sure why it is disappeared although you are a SearchGPT waitlist user.
Maybe, it is being updated.

Also, you can read this topic how to know which tool is being used in conversation.