The Samsung Galaxy S5 is VoLTE compliant. It should work fine. That said, sometimes it has to be in 'global' mode to make/receive calls. My S4 Mini will send/receive texts but not make/receive calls unless it's in Global (and not LTE/CDMA/GSM) mode for some reason. Which is kinda odd since if I'm in global mode, my phone randomly goes into 'searching for service' and I have to use airplane mode on/off every hour or two to restore it. I keep mine in LTE/GSM/CDMA mode since I don't use my phone to make calls, just send and receive texts.
If the S4 Mini is able to work, the S5 should most definitely. It's possible something is amiss on the carrier end, as sometimes I've had to phone in Verizon (my carrier) to fix issues that end up being on their end. Once, when I was still using my Thunderbolt, I was unable to send or receive texts one day (SMS code error cause 2 whatever that means) and they somehow messed up on their end and had lost my IMEI info, which they promptly restored.
It's also possible you're dealing with a service outage. Sometimes this can last for days, and it makes your phone act like your account got deactivated. Also, sometimes the S5 is buggered and needs a reboot (it was not the most reliable phone I've owned).
Even if your phone wasn't compliant, you should have received text messages indicating you need to upgrade to keep the service. If you haven't gotten any, then it's likely something else.