The radio is not "on" all the time. Just as in any other cellphone, it makes very brief contacts to the towers to update its registration with the network; this consumes a small amount of power, but the power management design of the radio and the coprocessor which runs it are pretty well done, so it can power up the circuity it needs just long enough and then power it back down again. Without that we wouldn't be getting multi-day battery life even from simple candy bar phones.
The screen in a smartphone itself consumes some power, but any any time the screen is on, the application processor - which runs linux and android - is also going to be on, and that consumes a lot of power compared to keeping the radio processor registered on the network but not actively exchanging data. With the screen off and nothing holding wakelocks, the application processor can try to go to sleep, and only wake up every few minutes to make quick checks. That's where your real power savings comes from, and yes, scheduling more or less frequent email polling can change this.