Jetpack not connecting despite WordPress.com dashboard saying it has

I click the Return to your site button (which btw way needs renaming because that is false) which sends me to the Jetpack plans page (not my site see?).

That makes sense! I took note of this here, and we’ll see about updating this link.

Clicking Free at last sends me back to my Dashboard but the Connect Jetpack button is still displayed (refreshing the page makes no difference either).

Do you use a caching plugin or service on your site? If so, could you try clear your site’s cache?
Once you’ve done so, go to Plugins menu in your dashboard and deactivate, then delete the Jetpack plugin.

You can then reinstall Jetpack from scratch and try to connect again.

If you don’t have a caching feature, or you aren’t sure where to find it, we have a little alternative connection route plugin you can use.

Can you download this plugin:
https://github.com/automattic/jetpack-addons/archive/add/add/old-connection-flow.zip

Download the file, then go to Plugins > Add New > Upload in your dashboard to install and activate the plugin. Could you then try to connect again to WordPress.com? Once you’re connected, you can deactivate and delete this little plugin.

Let me know how it goes!