I have been using phosphate-free dishwasher detergent since long before I campaigned for the reduced phosphate law we now have, so I know it works. What I also know is that machines need maintenance and that people frequently neglect that. Large amounts of harsh phosphates do help to make up for clogged screens or sprayers, but the fix is easy.
All the dishwasher machines that I’ve looked in have a lock cap on the sprayer arm that removes easily. Then the arm and strainer screen can slip off to be washed in the sink. Use a scrub brush on the screen, then check the sprayer arm to see if its holes are clogged. If the holes are clogged the arm may need to be opened by removing clip locks from the ends. A similar spray arm will be found on the top of most machines. At each stage be careful to watch for washers or gaskets and take care to reassemble in proper order. Our family’s machine needs this done once every couple years.