An inch proud of the cabinet faces is too much. Ours sat out a little when I first slid it in, and getting it flush took maybe twenty minutes of fiddling, so it's definitely fixable in most kitchens. First thing to check: the shipping brace. A lot of units come with a plastic or cardboard piece bolted across the top rear of the tub to keep everything aligned in transit. If the installer left it on, it can hold the machine off the wall by an inch or more. Pull the dishwasher out a bit and look across the top back. The install manual usually mentions removing it if you need the clearance. Second, look at what's behind the unit. The junction box where the power connects, the water supply line, and the drain hose all live back there, and if they're routed badly they'll keep the tub from sitting back. The drain hose especially loves to bunch up behind the machine. You want those lines to run off to the side or down low, not sit straight behind the middle of the tub. Third, the counter overhang. Most counters hang over the cabinet face by roughly an inch, sometimes more. The dishwasher body itself usually fits under fine, but the door is thicker than the tub, so the door is what has to clear the overhang. If the feet are cranked up too high at the front, the top of the door can catch on the counter and stop the machine short. Lowering the front feet, then leveling side to side with a level on the door rack, usually gets everything tucked in and lets the door sit even with the cabinet frames. The adjustable feet you read about are real. They're at the front bottom corners and you turn them by hand or with a screwdriver to raise and lower the unit. That's for leveling and for getting the door height to match the cabinets, not for pulling the machine back, so if it's sitting out an inch, the cause is behind it, not under it.
As for the inspector, they only check code stuff like the drain loop and electrical connections. Flush with the cabinets isn't on their list. Call the installer back and ask them to pull it out and see what's stopping it from going home. If they just set it in place without checking, that's on them, not your cabinets.