Hi Hojo,
Since Copilot Chat still works on mobile and you can continue existing conversations, this does not look like a general account failure. Microsoft has recently updated the Copilot experience and introduced Copilot Search as a separate experience from Copilot Chat.
On the web, try opening Copilot Chat directly rather than the Search page. Microsoft currently documents the Chat entry point at m365.cloud.microsoft/chat. If Chat is available there, you can also pin Copilot Chat again in the Microsoft Copilot app.
If the direct Chat page also removes the New Chat option while the same account continues working normally on mobile, I would report it through Microsoft Support/Feedback as a web and desktop account-experience issue, since reinstalling and clearing the browser cache would be unlikely to resolve an account-side rollout problem.
Thank you, and feel free to respond back for more assistance.