If all you need is an extender, any Xbox that boots will work. Banned Xbox Slims on craigslist would be perfect, or ones with broken DVD drives - should be able to get one of those for less than $100. All Xbox360 models will work as extenders but the Slims are the ones to get because they are quieter and have less power draw.
Don't go wireless, way too many variables. Some get it working good all the time, others don't. Go with MoCA or powerline adapters if you can't run ethernet.
Using an Xbox is just like sitting in front of your HTPC, you can schedule recordings, watch live or recorded, see the guide, etc.