Actually, you can also with Migadu, for free. You'll just get a "Sent via Migadu" signature on the bottom. And almost free without the signature :)

However, if you need to send thousands of messages a day, you are probably sending automated mails, transactional or newsletters. Migadu is not intended for that, just as Gmail isn't.

You don't need Mailgun for that either, you can setup your own server to send mails on DO, or go for Amazon SES (Simple Email Service), cost is negligible.