I fixed this myself using my S4 settings as a template. Here is how you can fix it:
It was the APN. See below in bold for the settings that were not right.
If settings not correct, must create new profile. Here are correct settings:
Name: Default is T-Mobile US LTE, but new profile can be anything, say T-Mobile
APN: fast.t-mobile.com
Proxy: Not Set
Port: Not Set
Username: Not Set
Password: Not Set
Server: Not Set
MMSC: http://mms.msg.eng.t-mobile.com/mms/wapenc
Multimedia Message Proxy: Not Set
Multimedia Message Port: Not Set
MCC: 310
MNC: 260
Authentication type: None
APN Type: default,mms,supl
APN Protocol: IPv4
APN roaming protocol: IPv4
Turn APN on/off: (checked) APN is on
Bearer: unspecified
Mobile Virtual network operator t...: None
Mobile Virtual network operator v...: None
In both phones where I had issues, the settings came from the factory as:
APN Protocol: IPv4/IPv6
APN roaming protocol: IPv4/IPv6
Note: one of the above protocols might have been IPv4 - not looking at my wife's settings so I can't remember, but one was definitely IPv4/IPv6. Looking at my S4 right now, they are both set to IPv4.
Not being a phone expert, I don't know which of these or if all of these created the issue. I just know that as corrected, it works and I have no issues of reduced functionality on my S4 now for 8 months. Now that I found it, I am CERTAIN that this was the EXACT same issue with my S4.
As set from the factory, wifi calling will connect for less than a second, but immediately drop. The phone pops up a message that says "You've successfully completed your first wifi call". Of course, it's not very successful if it only last for a fraction of a second. However, this message would indicate that the hardware is working and that it is a settings issue. The user can check a box to not see this message again after the first one, so the lack of this message may not mean the settings are correct if the message has been disabled by the user.
Need to work on the software or the factory setup process to eliminate the glitch as well as troubleshooting this issue quickly.