Hi and welcome to the forum Lars. I'm not ignoring you but I do very little die cutting AND am in the UK so not really in a position to offer assistance on this front - pretty sure you SHOULD be successful for less than $600 and hope you will share information once you've made your choice.
I'm not aware of any cutting device in NA that would meet those specs. At least, not those machines that are marketed for the craft market - there may be a more industrial type machine that fits those specs, but it is not an area I know much about. There are certainly many machines out there under the $600 range, but all the ones I am aware of would fail in at least one of your criteria areas. A cricut you can sort of automate, but you can definitely only feed in one sheet of paper at a time. With the right kind of die, it is possible to cut multiple sheets of paper on a big shot, but that is definitely manually operated and not automated. Digital cutting machines that work with your computer, such as the silhouette line, brother scan 'n cut & others like it may fall closer to what you are looking for, in that you can automate the cuts somewhat with the computer. But as far as I know they do not allow for loading of multiple sheets of paper at a time. Here is a review article of some of the more common ones of that type. www.toptenreviews.com/home/crafts-sewing/best-die-cutting-machines/
WIP Elegant Irises by Amanda Butler Lazy Days by Mystic Stitch pg 1/24 (BAP) Nativity Story Table Topper pg 1/6 by Blackberry Lane Designs (BAP) Amazing Grace by MBT (BAP) pg 1/6 Baby Ark Sampler by Bothy Threads pg 1/4 Skate by AAN Herbes et Epices by Madame La Fee Ornaments 1/12
Completed 2020 Names of God by MBT Ponchos x3 (2 nieces, 1 doll version!) Christmas ornaments - 11/11 Soffia by AAN
Just wondering if any one on chat would be interested in a large assortment of card making accessories. All new and unused. Printers inks pens in fact to much to list but can and will send anyone interested a list with pictures of what I have Enough to hold a small market stall and lots of variety. My name is Lisa and I lived in Leicester