Charise, thanks for your reply.
Automatic network switching has never been enabled on my phone for the reasons I have already cited- I insist on having control over my Mobile Data toggle to avoid paying for data overages. The Mobile Data switch used to be clearly visible in the drop-down settings, until an Android update took that away, which I always deplored as an underhanded strategy by cellular providers to force customers into data overages by concealing the Mobile Data switch several layers deep in a menu system. Underhanded tactics like this are just another reason that people distrust and dislike cellular providers today. So I installed a widget to restore my ability to quickly and easily switch Mobile Data on & off.
I keep Mobile Data turned off at all times to avoid data overage charges. When I want to receive or send an MMS while at home and connected to wi-fi, I simply turn on Mobile Data until the message transaction is done, then turn it back off. Again, as I have stated clearly, it always worked before the 4.4.4 update. Also, two different AT&T reps have assured me that MMS works fine over wi-fi, that the only reason Mobile Data has to be turned on to send/receive MMS is that the message header info has to come over the cellular data network, but that the MMS message itself, including the image(s) or video(s), passes over wi-fi and the entire process does NOT run my data plan usage up.
I am fascinated by all the replies here alleging that MMS over wi-fi is just not possible, yet nonetheless AT&T Wireless Tech Support spent hours in chat with me 2 days ago trying to make it work like it always worked before the 4.4.4 update, and now sweet Charise, an AT&T employee, is posting suggestions to try to help me get it working again. Why would all of these AT&T employees do that if it was impossible for MMS to work over wi-fi? Are those of you who claim it's impossible actually saying that you know this to be a fact, and AT&T Wireless Tech Support (and now, Charise) are so un-knowledgeable that they don't know what you know? Also, my preferred messaging client, Textra, has a setting to "prefer wi-fi for MMS when available." Why would this program have a setting to enable something that is impossible???
As info, I have tried the AT&T Messaging app as well as the other (default-generic) messaging app that came pre-loaded on the phone (it's just called "Messaging" so I don't know if it's a Samsung app or an Android/Google app). With all 3 clients, the result is the same: although I used to be able to, now I can no longer send or receive MMS over wi-fi, or even when wi-fi is turned on.
Thanks for trying to help, Charise, but Auto Network Switching is not the culprit because it is, and always has been even when MMS over wi-fi worked for me, disabled.