Where did you get the idea that most Amish spurn refrigeration? Many don't have refrigerators, but that doesn't mean they don't have refrigeration. Those without refrigerators usually use ice boxes.

I don't think most Amish view the issue of being tied to the outside world or not as a binary thing. It's a matter of how tied they wish to be. Using diesel and using electricity from the grid both tie one to the outside world, but not to the same degree.

With diesel you have many possible suppliers so if you have a problem with one you can switch to another. You can stockpile it so you can go for a long time without outside contact if you need to.

With electricity from the grid you generally only have one supplier. When the grid goes down you are down. You have to physically connect to the grid, and have equipment owned by the power company on or near your property, such to visits from the power company.

All in all, going on the grid entwines you much more with the outside world than does using diesel.