Allow me the privilege of answering my own question, which might be useful for others in future. And then, a few comments...
My setup might be a little unusual to begin with, where I treat the first user-profile I log into upon boot as a high-privilege user (the one in-charge of Settings for the device). My daily driver is another user-profile. In investigating this issue, I learned that the Messaging app is unable to interact with the user if of the 'another user-profile'-kind!
Two such situations are when...
(1) MMS has a 'Tap to Download' link (not sure if this mechanism is carrier dependent)
This link is displayed in the Messaging app only in my first user-profile. The MMS image/multimedia file will be downloaded if clicked. However, the same MMS, when viewed in my 'another user-profile', shows a 'small, blank-text, green-background box' with no way to access the image/multimedia file! One can 'long-touch' that box and select the get-info icon to see the "Message Details", but thats about it.
(2) If, for some reason, a text I am sending out requires me to 'Confirm' that that text might result in a charge and that I am OK with it. Again here, this works as required when in my first user-profile but shows "Sending..." and results in "Failed to send" or "Try again" when in my 'another user-profile' because it didn't get a confirmation.
For now, these are interesting quirks and I at least know how to get around it - access MMS and confirm sending of text when in my first user-profile. A small hassle for sure but will live with it. I would appreciate if there is a better way (i.e. Messaging app behaves the same in all user-profiles).
Now for the comments...
Firstly, thanks Mr. Hulk, I appreciate your response. I understand the dangers in SMS/MMS that you have rightly highlighted. Sadly, it is near impossible to stay away from SMS/MMS since it is the lowest common denominator. The other day, a friend of mine sent me a pic of a book cover and I had to request him to text me the title instead. He is friend enough to laugh it off that my MMS doesn't work and texted me the title, but he isn't friend enough that I could get myself to try and convince him to use Signal always!
But yeah... thanks tho. Take care.