Hi
As it happens I am thinking of getting a NUC for oter purposes.
I came across this one for $300, ready to go. For your purposes, may need an internal drive and more RAM, adding to the price, but, at least it has an operating system and is otherwise ready to go.
Power required is 19V DC, but, this is likely to still be an issue.
As for no monitor, you are correct, but, I believe that these do have normal video output, that would be readily seen with Teamviewer etc.
Overall, I suspect that some of the other options would be better for your purpose.
https://www.scorptec.com.au/product/...-BOXNUC6CAYSAJ