We want to leave that whole thing out of the checkout process, but leaving shipping method unselected makes the page unable to proceed to final checkout and place the order, throwing an error message on the page just above the delivery methods section: "Please select a delivery method". It does not scroll up to this message automatically, so the user is left wondering why the next page won't load after clicking "Continue" at the bottom.

Yes. You can have an extension set the shipping method for the customer and hide the selection module from the view

But in order to have even one Shipping Method available to select whether programmatically or by the user, I have to turn on Charge For Shipping in the Accounting Preferences. We want to turn that off. I'm trying to find a solution where Charge For Shipping is turned off, and shoppers can still Place Orders.

@Aaron McCausland on the website set up record > Shopping subtab do you have “Shipping information is Required” checked or unchecked ?

I just noticed that checkbox. It's checked! I'm going to try unchecking it.

If you ever wanted to be able to support both items that need shipping and those that don't, I wanted to note that Gift Certificates behave the way you want, even with that checkbox checked. They skip the shipping page of checkout completely -- at least in the versions of checkout I have fiddled with (Checkout 2.0.4, 2.0.5 and Kilimanjaro). There was a "patch" to allow the behavior. I would perhaps look into finding the hook in the code that exempts Gift Certificates from shipping during checkout, so you could add more exceptions as needed. Checkout Suite Answer # 46484 to see the patch to 2.0.4 and 2.0.5. It may give you clues as to where you can make a quick surgery. If you never need to ship or collect shipping, then unchecking "Shipping Information is Required" does sound easy/ideal.