For something like this, I would just run Debian. (Use whatever DE you're comfortable with)

Mount the NAS share through fstab and install mpv as a media player.

Well if you are familiar with servers I'd can't stop myself from shilling the setup i use, you install mint, then you set up i3 as the window manager, 500mb of ram on idle, I set it up in a 4gb ram laptop and runs beautifully, stable and never breaks too.

How much work do you want to do?

None - Librelec is already configured for media center usage. Install and use.

Little - Fedora IOT. Set up the system, which has a small core, verify you've got the right drivers, plasma bigscreen, your media apps.

Lots - Arch linux. Same thing, set up the small core (manually), then handle drivers, plasma bigscreen, your media apps.

Lots (But I don't like systemd). Go Void. It's the same as arch (more or less) but you use runit instead of systemd.

Full Control. Go Gentoo. This is probably overkill here unless you've got some kind of restrictions to meet. You know your needs best.

I have a super old lenovo nuc that I use for the same purpose. I ended up putting Linux mint with xfce on it and it works great. Not the fastest for sure but def usable

Everyone is going to recommend their distro :P.

My new home is openSUSE tumbleweed. So you can have a stable hybrid server/gaming desktop :)

Use Bazzite. It's unbreakable and comes with lots of easy installable packages. And I say that as a looong-time Debian fan.