Again, how is such feedback helping the OP? It's like you are committed to confusing the point... being, I'm pretty sure what he is looking for is highest quality playback (1080i+) OF VIDEO. I don't think he is asking about the resolution of menus nor will it upconvert lower quality video source to higher quality video resolution.
He's looking for a way to capture 1080p in a DVR-like solution. If he finds a way to capture that video, he is also looking for a way to play that video back. Answers like yours encourage potential confusion. For example, this answer makes it sound like upscaling could be equivalent to 1080i or 1080p. It is not.
The facts remain the same: if he's interested in watching 1080i/1080p content and the choices are either Mac Mini or

TV, his ONLY choice is Mac Mini.
If he also wants DVR-like HD capture, the ONLY way that's going to work is with Elgato-like devices hooked either to a master computer or that Mac Mini. The

TV has no DVR functionality, nor DVR add on options.
I have an

TV and I like it just fine. But clearly if you read what he is seeking, there is only ONE choice for his need. Throwing out "answers" that could breed confusion for the OP is not helpful. If YOU have an

TV and you are happy with it (like me), great. But his need is for things beyond what your & my

TV can handle. We're not putting our own

TVs down by pointing him toward a Mac Mini that can actually cover what he needs.
So, one more time:
-If he wants 1080i/1080p video quality playing back, Mac Mini can do it,

TV can NOT do it.
-If he wants 1080i/1080p HD DVR functionality, Mac Mini with an Elgato like add on can do it,

TV can NOT do it.
And if I've misread his post and he wants 1080i or 1080p menus, then

TV can manage that. Upscaling of
video was news to me, but even if that is true, upscaling lower res video does not result in real 1080i or 1080p video playback, just scaled-up lower-res video.