Well Google TTS is a method not an app, per se. It is an app as is everything on the device but many messaging apps accept commands from TTS (text to speech). TTS also falls under Google Assistant as GA is what actually acts on the commands. We don't need to over complicate things.
I'm looking for the name of the app that sends the message. Find your messaging app and tell me what the name under it says. I mean you could be using Google Messages, which would probably not be a native app, these are known as 3rd party apps. It would be something you downloaded from the Play Store. You could also think of native apps as default apps. You opened the box and started using the phone just the way it came. They are sometimes referred to as pre-installed apps. They often have names like Messages, SMS or Text.
The big reason I'm asking this is because I've seen cases in the past where uninstalling the downloaded app and reinstalling it has cured issues like this. Something just gets stuck and clearing caches and data just isn't enough. The problem with that solution is native/pre-installed apps cannot be uninstalled so that would leave that option out. You might be able to accomplish something similar by installing another app, transferring everything to it, making it your default then trying to switch back to old app if you really liked it that much. A lot of times people switch because other apps have better features or better looks, they might be easier to use. It's all personal taste though.
So if you can uninstall the app. The best way to do this would be to figure out the name of the app, then search for it on the Play Store. When the page comes up it will say Installed, and there either will or won't be an uninstall option. If there is, just uninstall it and without leaving the page install it again. Set it as your default and try it out. If it cannot be uninstalled find an app with a good rating, 3.7 or higher, closer to 5 the better, install it. Make it your default move everything to it, test it, then move back if that's what you want to do. Some pretty good SMS apps, Next SMS, Messages+ (by Verizon), Google Messages, Go SMS to name a few.
Post back what you find and how it went.