Well, you are partially right. It is not a carrier issue nothing updated there but, I took the information you provided and applied it to the iPhone, and it worked. I mean I sent a huge file just to test it too.
These instructions worked for my iPhone 14 after updating to iOS 16.6
Step 1. Go to Settings.
Step 2. Scroll down to Messages and press the '>'
Step 3. Scroll down to MMS. If it is Off, turn it On, if it is On, turn it Off (Mine was ON already).
Step 4. Say "Hey Siri, Reboot" and press the restart button.
Step 5. Repeat Steps 1, 2, and 3. But this time making sure the end result is MMS switched to on.
Step 6. Repeat Step 4
Sucks to have to go through all of that, but makes kinda sense I guess, Apple care is always having me switch the MacBook Pro , iPhone, and Apple Watch off and on so many times I think I will need a new on/off button. I like telling Siri to turn off though LOL. Hope this helps!