Generally speaking with Intel CPUs, the T or S models offer very little benefit because they're not much better (if anything) at idle and they tend to take longer to complete tasks which negates the lower power draw (obviously it depends if it is a continuous task like gaming, or a finite workload). Ironically, K models with turbo turned off tend to fall quite comfortably in the highest efficiency point of the architecture, as do the lowest models like the 10400.
Without a GPU, the motherboard usually makes a bigger difference to idle power than the CPU. ITX motherboards are usually better than other formats and ASRock tend to be the best mainstream brand for M-ATX, but it varies a lot. Intel and Fujitsu are historically excellent.
GPUs add quite a lot to idle power, unfortunately (more than just the card itself, because they prevent the CPU reaching lower idle states, unless something changed recently that I'm not aware of). The RX 6500 and 6600 cards are excellent, as is any Pascal card. The 1660 Super isn't bad, so probably not worth upgrading. High refresh rate monitors and multi-monitor setups can significantly increase idle power, AMD historically have been much worse than nvidia for this (Polaris was pretty bad, up to 30 odd watts at idle!), but I don't know if that has continued with current gen.
With a good choice of components, getting sub 30 watt idle is fairly easy with Intel, but lower than that needs you to be pickier about the GPU and motherboard. Less than 20 usually means no IGP and less than 15 a highly efficient, probably a pico-type PSU. OEMs can often offer lower idle than a typical desktop setup / PSU because they're simplified.
From what I'm aware of, NAS forums are a good place to go for advice and there's a long running thread here:
https://gathering.tweakers.net/forum/list_messages/2096876/10
I'm not aware of much specific advice on 11th or 12h gen. 8th/9th gen has really established boards and setups and the performance increase of 10th/11th wasn't really worth it for desktops or NAS. Kontron are supposed to be releasing 12th gen boards in Q2 and they've said they will prioritise power consumption like the 8th/9th gen Fujitsu boards did.
