I was under the impression WiFi Calling was just regular phone service through WiFi. It seems to work that way for me, 2FA codes and all.
WiFi calling seems to depend on your mobile service provider to provide the service. I guess different networks do it differently.
VoWiFi (as Wi-Fi calling is called in the 3GPP specs) is similar to VoLTE, but not all SMS go over VoLTE: Unlike for calls, where there's mandatory VoIP in 4G/LTE and beyond (there is no more circuit switching), there's still a fallback path for SMS that uses legacy signalling instead of IMS (which powers VoWiFi and VoLTE/VoNR).
Maybe there are some SMS gateways that are somehow incompatible with some IMS message gateways? (Theoretically, the IM-SM-GW should be transparent to external networks, I believe, but practically I wouldn't be surprised if some weird things lurked in there, requiring a fallback to the signalling path, which is not available on VoWiFi.)