Cookie Consent Block won't hide / Incompatible with cookie-unaware caching

### Impacted plugin

Jetpack

### Quick summary

The Cookie Consent Block doesn't stay hidden when the website uses a non-cookie-aware caching system; or where it is aware of cookies, but doesn't differentiate upon the `eucookielaw` cookie used by the block.

This is because the block only hides itself **server side**. If a cached page is served to a browser, which includes the Block html, the prompt will be shown again.

The block should check for the cookie **client side** and auto-hide the block at display time.

### Steps to reproduce

1. Install WordPress & Jetpack on a server with `eucookielaw` cookie-unaware caching.
2. Dismiss popup
3. Reload and view popup again

https://github.com/Automattic/jetpack/assets/767313/58284bb1-39e8-46c0-92bb-e290e5d414b4

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

_No response_

### What actually happened

_No response_

### Impact

Some (< 50%)

### Available workarounds?

No and the platform is unusable

### Platform (Simple and/or Atomic)

_No response_

### Logs or notes

_No response_

Start by locating the Jetpack Cookie Consent Block implementation and its handling of the eucookielaw cookie. Reproduce the reload behavior with cookie-unaware caching, then verify that a dismissed prompt remains hidden when the cached page includes the block HTML.

Written by the indexing model from the issue text.