Your first email API usage help (after selecting domain and API option, the one with header "How would you like to send your emails from?") does not contain very important note from https://documentation.mailgun.com/en/latest/quickstart-sending.html#send-with-smtp-or-api:

NOTE: If you’re sending from our EU infrastructure, be sure to substitute the beginning of the endpoint “https://api.mailgun.net” with “https://api.eu.mailgun.net”
So if you selected EU domain you will not be able to send an email using only that help. Either note should be added there or better API base URL should be corrected.