The old first generation LTE chips weren't the best at good battery life. Also the media ran with it because Verizon was one of the first carriers to have LTE but their phones ran TWO radios at the same time (both LTE and CDMA) which would destroy certain phones battery and so people just assumed that LTE causes bad battery life.
Now with the newer generation LTE chips they save battery life because lets say you go to a website in Safari it loads so much quicker on LTE and so your phone can go back to a low power state, whereas with 3G you might be loading a page 5-6 seconds longer. Over time it adds up.