Fit is only the first filter. Even a pan that clears the top spray arm has to actually come clean, and that depends far more on what's stuck to it than on the machine's program list.
A rinsed saucepan — light residue, no burnt-on layer, run under the tap before loading — is the realistic best case for any of these machines. Spray arms move hot water under pressure with detergent, which handles surface grease and food residue about as well as scrubbing by hand, sometimes better on stubborn spots a sponge glides over. The EUHOMY and AIRMSEN countertop units both run multi-program cycles designed around exactly this kind of load: dishes that need a real wash, not a hose-down.
A baked-on roasting pan, a skillet with browned fond stuck to the bottom, or a pot that sat with dried oatmeal in it overnight is a different category of problem, and it's the same problem for a full-size built-in dishwasher, not just a compact one. Dishwashers of any size clean by spraying, not by soaking or scrubbing; a layer that needs mechanical abrasion or a long soak to loosen won't dissolve just because the water is hot. A single short program in a compact machine, with a smaller water reservoir and a shorter cycle than most full-size dishwashers run, starts from a weaker position than a large built-in on a Heavy cycle — and even that often still needs a pre-soak.
The practical rule: scrape and rinse before loading, every time, in any of these machines. It's the same advice full-size dishwasher manuals give, just more load-bearing here because there's less water and less time to compensate for skipping it. Cast iron is its own separate case and belongs nowhere near a dishwasher of any size — detergent and an extended wet cycle strip the seasoning that keeps it from rusting, a manufacturer-documented reason to avoid it.