That's a bit of a simplification. Because VoLTE is still being standardized, phones must keep multiple modems active and multiple receive portions of the transceiver ASIC powered on in order to be able to receive voice calls when on LTE. This can be mitigated some on UMTS, because I believe a LTE device can get paged over to WCDMA when a voice call comes in through CSFB, but that cannot be done with the current deployment in traditionally CDMA networks. There was a proposed standard for CSFB to CDMA 1x voice, but I don't think it was adopted by any of the carriers due to some concern with call setup delays and delays to being able to deploy LTE.
Aside from that, you're perfectly right about how power-hungry the current LTE implementations are, because they're early implementations. They will get better as time goes on, within reason.
Another thing that's probably hurting battery life right now is the relatively large size of LTE cells. While coverage is pretty good, if you're a long way from a cell tower your mobile has to transmit at a higher power.
Aside from that, you're perfectly right about how power-hungry the current LTE implementations are, because they're early implementations. They will get better as time goes on, within reason.