My main concern with these higher frequency connections is that they require a lot more power to run. So we can’t have phones that will last weeks without charging.
That's generally not true.
What does require more power is a higher data rate (which is often available only on higher frequency bands, due to there being more bandwidth), but there shouldn't be any meaningful difference for a constant data rate voice call across bands.
If anything, higher frequencies usually imply a denser network of base stations (since radio propagation is somewhat worse), which means your phone needs to transmit less in the end since there's less distance to cover.
For example, GSM at 900 MHz had a maximum mobile EIRP of 2W, while at 1800 MHz it was only 1W (at the same channel bandwidth).