In US hotels, what happens if you skip checkout and just leave the hotel?

At many hotels, particularly those affiliated with major chains, it's perfectly fine. It's common for such hotels to leave a copy of your bill under your door sometime during your last night (they'll later add any charges you incur in the morning, of course) with a note that you can simply accept the charges to your card on file by leaving. There's usually a box by the front desk to drop off the keys so they can be reused. In some cases, there's an option to "check out" on the TV's interactive menu or the phone system, which lets them know you're done with the room. Many hotels now can simply email you a copy of the bill as well.

For example, this express check-out feature is expressly stated as a member benefit of Hilton's rewards program.

If your hotel doesn't offer an express checkout scheme, I would find it rude to leave without checking out, yes. They need to know when you're out of the room for housekeeping purposes. If you just disappear, the staff may take extra time to contact you with a copy of your bill. And while I've never had it happen to me, it's always possible some hotel manager will accuse you of staying past check-out time because you never checked out, and that's a bunch of trouble you don't want.

If you aren't sure about a particular hotel's policy, you can always ask when it's not busy. Even if the hotel doesn't have a formal express checkout policy, when I've known I'd want to leave in a hurry in the morning, I've asked to "settle the bill" the night before, and they've arranged things to expedite my departure.

Housekeeping reports back to the main desk when a room is empty (back in the 80s when I was a desk clerk there was an automated system using the room phone), and it gets rented again. Whether or not you checked out isn't actually the trigger. Checking out puts the room on the cleaning list, it doesn't put the room on the rent-me list.

And in the mornings, by the time there's a queue at the desk for checkout (usually populated by tourists who rarely travel, not business travelers), housekeeping is working at full capacity, so actually checking out doesn't really matter. If you're checking out very early, then they'll probably clean the room early too, but then there's a clerk at the desk with no other guests to deal with.

At the hotel I worked at, the key point was around 3pm; housekeeping was finishing up, and then they'd have to check the rooms that were scheduled to check out that day, but housekeeping had reported occupied + cleaned before checkout time. If they were empty, they got a fresh clean (pretty cursory) and then went on the rent-me list. (The only time you'd get charged for a late checkout was if you actually checked out at the front desk after checkout time. If you just left, we'd never know.)

You will simply get charged the pre-agreed amounts on the credit card you gave them when booking.

Many frequent-travellers don't check in or out - you get your electronic room key in the app, and just leave when you are done. All major chains support something similar; I haven't visited any check-in/out desk for years.

One advantage of checking out at the counter is that it gives you an opportunity to receive a paper invoice and verify on the spot that the amount charged by the hotel corresponds to what you were supposed to pay. Mistakes do happen -- once a hotel had added a $50 "valet parking" charge even though I don't own a car...

Probably it is still possible to dispute such charges after you have left the hotel and they have charged you for the stay, but this is more complicated than fixing the error at checkout -- especially if you live abroad in a country where you can't easily dispute credit card charges.

This practice is very common in the US, since a credit card is taken at registration. However, I have family members who, accustomed to US practices, attempted to pay at registration at a hotel in Scotland. They were not allowed to do this; they then forgot to checkout the next morning. They spent a night in jail in Scotland before things were sorted out!