I'm not sure why you are crying no server and at the same time asking for a 1U solution. Makes absolutely no sense because anything in a 1U format is going to be called a server even if in reality it isn't. Your obsession with the word server is blocking you from finding the right solution for your needs. Noise is controlled by buying the coolest running processor that will do the job. That means a 35 watt processor instead of a 90 or 120 watt processor.
For example this board seems to meet a lot of your needs:
https://www.newegg.com/gigabyte-b450-i-aorus-pro-wifi/p/N82E16813145086?cm_sp=SearchSuccess-_-INFOCARD-_-Mini+itx-_-9SIA4REBMZ9832-_-2&Description=Mini+itx. It is a mini-ITX and could be slotted into a Mini-ITX case from Supermicro, add some fans and you could have the perfect platform that is an expensive board but there are solutions well under $130. In the end you get what you pay for and frankly I'd consider something more upscale for a the better features. Right now AMD is the better option for performance per watt but if you have a couple of weeks I'd wait for the new APU's to come out and would look towards the lower end power wise.
So why this particular aorus? Nothing specific, it was the first one found. I does have dual video ports and an M.2 slot. You will want an integrated M2 slot on anything that gets knocked around in travel. However these days even $130 class board has an M.2 slot. the only negative with AM4 boards is the lack of a lot of 1U cooler options with only one listed. the other reason I'm looking at Mini-ITX is to find minimalist board that hopefully runs much cooler than an ATX board.
In a nut shell if you want a 1U rack space solution you need to look at 1U server chassis, compact motherboards, Mini_ITX in this case and APU type chips.
I said NO SERVER so it should get caught even before the "quiet".
There are numerous reasons why not a notebook. Dedicated rack PC with just PWR-ON button on the front is all that was needed (and couple specific ports on the back side, which servers do NOT have).