Old photo, but yes, it has a button on the side. 
The MiSTer costs around $500. Surely for $500 (the cost of a PS5), it comes with an on/off switch...
Discussion
Actually… now I think about it, mine only does because I have a BlisSTer Hub as part of my stack 😁
I'll have to look up BlisSTer. I wonder if it works with the MiSTerpis. At around $190, the MiSTerpis are interesting to me (if they ever stay in stock), but I saw everyone either yanking the cord or using an inline switch on their cable which is insane to me. $450 is a premium product, why is there no on/off switch.