I dislike FB Messenger and gmail because they're essentially spyware. EVERYTHING sent through them is parsed and analysed. It's how ads and even links on Google and FB are shown to you. People are too willing to give up basic privacy in favor of convenience and instant gratification. The tradeoff is that you give your online self away.
Services like tutanota and protonmail are built such that even the site operators can't read your messages. They also offer the ability to send an encrypted message to someone not on their service.If you're sending personal, intimate details over gmail and Messenger, you're essentially telling those details to thousands of people who have potential access to those messages. Using services like tutanota or protonmail is just putting an envelope over the stuff you might not want others to see.
If you have to use facebook, you can send a private message without using Messenger: make a private post (the "Only Me" option) and tag the person you're contacting. Messenger sucks, but sometimes for various reasons it may be the only way to get in contact with someone. But you don't want to have the awful Messenger app on your phone, so instead you can change your settings to make it a desktop site. In Android Chrome browser, tap the three vertical dots at the top right of the browser window and choose "Request desktop site". To revert to the mobile site, uncheck the "Request desktop site" box.