Unfortunately, your two desires are at cross purposes to each other. An external power brick helps keep a lot of heat out of the system, which allows you to run less cooling in the box, and thus be quieter.

Yeah, except Apple manages to come close so it's possible :)

Give credit where credit is due.

The mac mini is much larger than an Intel NUC.

There's two other factors that the Mac Mini doesn't do well on though: Performance (both per dollar and absolute), and size. Not having an external PSU is less important to me than both of those factors, and I'm sort of struggling to see how it could be considered more important.

The Gigabyte Brix [edit: the base model] is fanless, and if you use an SSD it's completely solid state. I have two next to me right now and they are completely silent as you would expect.

However it has a small external power brick. It's only about 1/4 the size of the machine (at least that's for the UK PSU).

Considering there are tens of options in the Brix lines the comment about them being fanless is untrue. Most are actively cooled machines. None of the Brix or NUC I have are particularly loud. If you're doing heavy lifting on a small form factor machine you have to expect a small fan to generate noise to dissipate heat, there's really no way around it.

I edited the comment to reflect the fact that the base model of the Gigabyte Brix -- which I have two of -- is fanless and passively cooled.

I've used a gigabyte Brix n2807 - it's slightly larger than other Brix models as it takes a 2.5" drive (using an old laptop drive to save costs). It's not powerful, but it runs Kodi and Windows 10 okay. Wi-Fi and Bluetooth don't work (or didn't when I first tried) with Linux out of the box. Up to 8gb LPDDR. No power brick. £80 on offer a couple of years ago.

Forgot to mention - fanless.

Well, I've checked the current offerings and I can't find any that doesn't have "19 V DC" as the power supply description. I guess I'm statistically irrelevant in their market.

Digging around behind the tv and it's a oversize plug and thin wire rather than a laptop style plug-cable-transformer-cable-computer set up. Not quite the mains-to-box set up you're after.

:)

Getting further off topic, the only other piece of hardware I know of where they bother is the Playstation. The XBoxes use ugly power bricks, just like the Wintel boxes.

The Xbox One S has an integrated power supply fortunately

I have a Skull Canyon Nuc and a Zotac EN1070 mini PC. The Zotac is quieter while gaming compared to the NUC and that surprised me.

Fan diameter is inversely proportional to speed, for the same volume of air movement.

Large, slow fans are much quieter (and easier on bearings!) than high RPM fans.

There are great fanless systems. I use a HD-Plex case with a real i7 CPU (although a few generations old by now) as a workstation. It's dead silent and with an internal PSU. It's been very reliable so far.