Hi,
About Elastic Email sending functionality, there must be something that you need to configure. From the image shared earlier, you are sending mails via SMTP/API. Make sure the ports are open. If closed, make sure they're open. If they are, Elastic Email provides alternative settings (ports/server) if your current configuration is not working.
Also make sure you have enough credits to send your mails and you have validated properly your account.
I hope this helps.