The power is part of why I prefer the embedded boards like the one I linked.
Especially putting one of these not in a rackmount chassis, you can opt for quiet cooling and fans that do not draw 10s of watts each. The idle loads can get fairly low, depending on your additional peripherals.
For very low power, I actually prefer small form factor machines, either used 1L SFF machines from the big OEMs (see STH's Project TinyMiniMicro[0]) or one of the many AliExpress mini PCs[1]. These sorts of machines tend to use low-power CPUs and a lot of the AliExpress ones are laptop parts.
As with all things, there are always tradeoffs. If you enjoy researching and bargain hunting for hardware, then this is a viable path. If you just want a machine that works, then this is onerous work.
[0] https://www.servethehome.com/introducing-project-tinyminimic...
[1] https://www.aliexpress.com/w/wholesale-mini-pc.html?catId=0&...