AI Chat block shows persistent prompt and error after enabling Jetpack Search

### Impacted plugin

Jetpack

### Quick summary

The prompt to enable Jetpack Search is persistent and there's an error when using the AI chat block on a site even after enabling Jetpack Search.

NOTE: This issue was found during the testing of p8oabR-1n7-p2


### Steps to reproduce

1. Add the AI chat blog to a site where Jetpack Search is not yet enabled.
2. Observe a prompt suggesting the need to enable Jetpack Search for indexing purposes.
3. Proceed to enable Jetpack Search.
4. Visit the page with the AI chat block.

### A clear and concise description of what you expected to happen.

After enabling Jetpack Search as prompted, I expected the prompt to disappear and the AI chat block to function without any errors.

### What actually happened

Despite enabling Jetpack Search, the prompt remained visible. When attempting to use the AI chat, I get the error `There was an error while generating the answer. Please try again.".

https://github.com/Automattic/jetpack/assets/67279475/62d2f1db-3736-4423-9355-7cc9587ced8e

### Impact

Some (< 50%)

### Available workarounds?

No but the platform is still usable

### Platform (Simple and/or Atomic)

Atomic

### Logs or notes

_No response_

Start by reproducing the listed steps on an Atomic site: enable Jetpack Search, revisit the page containing the AI chat block, and try to generate an answer. Done means the enablement prompt is gone and the AI chat no longer reports an error; the issue names no files or tests to guide further investigation.

Written by the indexing model from the issue text.