I also have a Mac Mini for this and agree the MBA alone isn't ideal for the reasons you mention.
But if you want the MBA and don't like plugging things in, you could get a refurb Apple TV for $75 and use Airplay to wirelessly send the sound and picture. Or if you have a PC that can run iTunes, you could use it for a server which would be available to an Apple TV and your MBA.
Picture quality with AirPlay is kinda crap. It's cool that it works at all but I don't think you'd want to watch it for very long.
I used my MBA as an HTPC for several months. I agree that it's inconvenient to unplug it when you want to go somewhere with it, and plug it back in later to watch TV. It's only 3 cables (power, video, USB hub) but that was still enough to make me reluctant to take the laptop anywhere. It was also annoying to have to constantly resize all my windows when switching between TV and laptop screens.
But, it's still doable. It works fine. These complaints are fairly minor.
In the end, I ended up building my own small custom PC to watch TV. Unfortunately there aren't very many fast, small, quiet, cheap PCs on the market for HTPC use. The ones that are available are generally pretty underpowered, using Atom processors or similar. If you know how to build your own (I built a mini-ITX system) I'd recommend that. You can set yourself up with a nice Haswell for around $300. As much as I like OS X, the operating system you use for an HTPC is fairly irrelevant. Most of your TV watching will be done in a web browser, and Chrome on Windows is basically the same as Chrome on a Mac.
If you are willing to spend an extra $100-$150 then you should be able to get a used Mac Mini (2011 or newer) and that would make a great HTPC. That way you don't have to worry about building your own, plus the Mini has a built in power supply (no need to use an ugly external brick) and the Mini is actually smaller than any mini-ITX system you might build yourself. And it's very quiet.