I don't really think the lack of a food grinder in European-style machines is all that big a deal, unless you just put in the dishes with all the leftovers still in the bowls or something.
With most European dishwashers, for example Bosch and Siemens, there's a large metal strainer at the bottom of the machine that occupies up to 1/3 of the surface. At the centre of this strainer there's a self-cleaning filter assembly.
That basically has a strainer that will stop anything larger than the drain pump can handle from entering. That leads to a filter chamber which is usually a cylinder of metal mesh, sometimes this may have multiple layers.
When the machine is washing, the wash pump sucks water through the sides of that cylinder assembly through the fine mesh filters and no small particles can recirculate back into the wash water.
When the machine empties the drain pump sucks water through the cylinder top-to-bottom which cleans the mesh filters and sucks any items away.
Small items of food e.g. peas, beans, pasta shells, rice, cornflakes etc will simply pass through the pump and down the drain.
Very large items e.g. slices of carrot, or maybe a tea bag you forgot to remove will sit on the top of the strainer and can just be removed by hand.
Cleaning the filter thoroughly is really very, very rarely required unless you're running very cold / short washes all the time or are using a really vast amount of heavy grease in your cooking or a very poor detergent.
