Google Messages “Not sent. Tap to try again.” on WiFi? A couple of things you can try
A few days ago I noticed that Messages, which is what Google Messages is calling itself these days, not to be confused with Samsung Messages which calls itself Messages, or… never mind I can list off a few… anyway, that program was no longer sending or receiving messages that were MMS when I was connected to WiFi.

The incredibly detailed error message was “Not sent. Tap to try again.” I could tap to my heart’s content when connected to WiFi and nothing. Always the same error.
Disconnect from WiFi and suddenly it was working again. Reboot the phone, no changes. Use a different messaging app and it worked fine on or off WiFi.
Now, I don’t send or receive a lot of pictures via MMS but I coordinate with several coworkers in MMS chat and having to disconnect from WiFi every time to download just a text message from a group, not particularly fun.
I did two things, the second one listed worked for a day, the first is what ended up fixing things for the past few days so we’re going with that as the best potential fix, although the second is less destructive.
Reset your network settings

I’d try this last if you use a lot of WiFi APs because, they’re going to be gone.
I don’t know why this worked, but it did. I lost all my saved WiFi passwords, but once again have what appears to be a stable Google Messages, even though every other Messages I have seemed to work.
Really not sure what’s up with that since they all use the same back end system API as I understand it.
Something to be aware of is you really should reboot your phone after you reset network settings. Weird stuff happened on mine where it appeared to work and then shut the LTE band off while I wasn’t looking and was down for four glorious hours in which I didn’t have to answer anyone’s texts.
Check data saver

While I was not in data saver mode, several days ago I toggled this to allow app while data saver on and it worked for a couple of days before deciding to not work again.
I’d try this first.
Unfortunately I don’t know exactly what the error was because Google’s not exactly giving me a very debuggable error.
What I do know is on the Messages app there were a lot of people complaining this week that the last update tanked it and needed work.