Wifi calling CAN work with texting. It's up to your carrier. Not all carriers, and not all cellphone plans allow wifi calling, and some charge for using it while roaming. You really have to find the correct answer for your particular plan and carrier.
It is not independent of your carrier at all. It just means that Tmobile, for example, allows your phone to access the Tmobile network through a wifi connection. Your text goes through the internet via wifi, to some Tmobile server, and then gets sent to your friend's phone in Chicago (using Tmobile).
A low cost MVNO may very well decide not to spend the money to provide that sort of service. Or they may choose to only allow voice calls (as the OP says). Or they may provide the service, but charge a few cents for each text or call if roaming. A while ago there was a thread here about Verizon charging someone a ton of money for making a lot of wifi calls from Europe. Maybe they had a really low cost Verizon plan and didn't know about add-on charges...
When I had an AT&T account, I used wifi calling in Europe to text friends back home for free. Now that I have Tmobile, I also text friends from overseas with wifi calling. A few minutes ago, I enabled Airplane mode, turned wifi back on, and texted my wife. No problem. In all cases, the text came from my "normal" cell number, and replies came back as normal, and I used my "normal" texting app.
If your phone's hardware is very old, it may not support wifi calling.
NOTE: The above stuff refers to "wifi calling" as selected on your celllphone's options. There are other ways to text or call, usually for free, using wifi, for example Google Voice, TextNow, Skype, TextFree, and many others. These apps do not involve your cellphone carrier, and the call or text does not come from your cellphone's phone number.