For all the effort (and money) put into making a media server as quiet as possible (which is still not silent), I think you're way further ahead to just not put the server in your living room. Instead, put it in the basement/closet/garage/utility room -- somewhere you won't hear it. Then, broadly speaking, you have three options:
A commercial box, like NVidia shield, Chromecast, Roku, Amazon Fire stick or AppleTV. This is by far the easiest as you get a remote, a "10 foot user interface" and they work with online services like Youtube, Netflix, etc. If you care about everything being open source software this obviously isn't a good option.
You can also DIY your own box on a Raspberry Pi. Be prepared to do lots of tinkering, though.
The other option is to get the media server's UI remotely. Long HDMI cable, HDMI-over-cat5 or wireless HDMI gets audio/video; some of those also do USB, or you can do wireless control if it's close enough. Almost as much tinkering as a Raspberry Pi, but could be cheaper if it's not far.
Another big benefit of the first two options is it scales to multiple clients, if you have more than one TV.
I can't recommend the long HDMI (or HDMI-over-CAT5, or long Thunderbolt) enough. The noise isolation you can achieve by doing so is unbeatable. Even moving anything short of a 1U server into a neighboring closet means no perceptible noise, so you're really buying yourself more flexibility such that you can use almost any computer without thinking about noise constraints.
Even more than that, I'll bet a lot of people who are connecting a media PC to their living room TV also have another PC somewhere in the house. With a long HDMI cable, you can connect an existing PC to the TV, thus saving the entire expense of a new PC. This is particularly interesting for gaming, since a good gaming PC is a lot more expensive than even some of the longest HDMI cables.
To add to the HDMI-over-CAT5 instead of long-run HDMI, I wanted to offer an anecdote: we tried doing HDMI with extenders at a previous MSP shop. We had random de-syncs. We were probably far past the 50ft spec for how long to run an HDMI cable and the extenders were unpowered (basically female-to-female boxes). HDMI-over-CAT5, however, handled the distance flawlessly without powered boosters in the middle.
I put a rack/cabinet in the laundry room, with a QNAP NAS in there, Plex server is running on the QNAP, then an Apple TV 4K connects to that and feeds the TV.
I like this better than trying to do some sort of long HDMI cable setup, because the HDMI ports on most NAS units out there get old rapidly, don't support HDR and high bitrates reliably, etc. This way, it doesn't matter if the server is brand-new and fast enough and all updated for the latest video standards. As long as it can push the data out fast enough via Ethernet, you're good. Updating an Apple TV every 3-5 years is a lot cheaper and easier than getting a new server every time. And a good NAS is a lot cheaper than a full-out server would be.
I have a similar setup myself. I use basically an older PC running Proxmox + openmediavault as my NAS (and a bunch of other things, including Plex). I currently have an NVidia Shield and 'Chromecast with Google TV' as clients, as well as a couple phones and tablets. It is pretty hands-off and my family can all use it just fine.
I'm content to replace the relatively cheap client boxes every few years. That said, I think I've had the Shield for over 5 years now and other than the original remote being bent in half (likely a jumping-on-the-couch-related failure) it's still going strong.
A commercial box, like NVidia shield, Chromecast, Roku, Amazon Fire stick or AppleTV. This is by far the easiest as you get a remote, a "10 foot user interface" and they work with online services like Youtube, Netflix, etc. If you care about everything being open source software this obviously isn't a good option.
You can also DIY your own box on a Raspberry Pi. Be prepared to do lots of tinkering, though.
The other option is to get the media server's UI remotely. Long HDMI cable, HDMI-over-cat5 or wireless HDMI gets audio/video; some of those also do USB, or you can do wireless control if it's close enough. Almost as much tinkering as a Raspberry Pi, but could be cheaper if it's not far.
Another big benefit of the first two options is it scales to multiple clients, if you have more than one TV.