Hi, I’ve solved it!
I’m explaining it in case it helps anyone else.
What happened to you is a very rare but very important case:
WordPress couldn’t connect to wordpress.org because the server had blocked the new WordPress IPs.
That explains why:
Rank Math failed when trying to check for updates or connect to its API.
You couldn’t install plugins (or see them in the repository).
Even after disabling it, the errors continued, because the problem was at the server’s network/firewall level.
Why does this happen?
Some hosting providers, for security reasons, filter outgoing connections and only allow connections to authorized IPs (a kind of “whitelist”).
If WordPress.org changes the IPs of its servers—something that happened recently, by the way—this filter prevents WordPress from connecting to its own ecosystem (plugins, updates, etc.).
How did I solve it?
I contacted the hosting support.
They identified new WordPress.org IP addresses that weren’t allowed.
They added them to the firewall’s whitelist.
Everything was back to normal!
Hello,
Thank you for contacting us. We apologize for any inconvenience this issue may have caused.
We are glad you were able to find a workaround to resolve the issue.
We have just released a Beta update that fixed the issue.
Please enable the Beta update by following this guide: https://rankmath.com/kb/version-control/#beta-updates

If you are unable to see an update, please clear Rank Math’s transients from WP Dashboard > Rank Math > Status & Tools > Database Tools > Remove Rank Math Transients > Remove transients

Please clear the caches, server, and WordPress after updating.
Let us know how it goes.
Thank you.