Proper way of permanently fixing a heavy utensils drawer with "hanger" rail?

If the drawer doesn't fall on the trip to the back limit, it would indicate that the plastic hanger is doing fine until it reaches the end of the support rod. Perhaps you'll find unusual wear on the far end of the rod.

As an option, consider to reverse the hanger clip so that it is on the inside of the drawer, shifting the hang point clear of the (possibly) failed rod end.

If you have a makerspace in your area, you may also find you can have a modified hanger created on a 3D printer. Using PETG (stronger than PLA or ABS), you could have a longer hanger clip (front to back) or a more narrow closure at the top to prevent the drop-out.

If you can find a second clip, attach one to the inside of the drawer and one to the outside, although that may not be as effective as a modified version.

Consider also to apply some wood paste wax to the rod surfaces to reduce the wear on the plastic.