A nascent standard called MHL (Mobile High-Definition Link) seeks to address these issues. It allows the TV to charge the mobile device over the same HDMI connection used to deliver video and other content to the set. The MHL spec (version 1.0 is already out) also lets you use your TV's remote to control playback on the connected device.
It might take a while for the technology to gain traction since both the mobile device and the HDTV must support it. But at least you wouldn't have to get a new TV: The MHL Consortium says you should be able to add MHL functionality to your TV through an short adapter cable that hooks into an HDMI port. As for phones and mobile device support, MHL Consortium members include Nokia, Samsung, Silicon Image, Sony, and Toshiba, so that's a decent start in the manufacturing community.
I hope MHL catches on: I hate crouching by my TV.
This story, "A Better Way to Put Phone Video on TVs" was originally published by Technologizer.