I ended up removing every reference to "ExtensionlessUrl" from the web.config and instead adding these two lines only

This seems to have done the trick.

Trying again:

[remove name="ExtensionlessUrl-Integrated-4.0" /]
[add name="ExtensionlessUrl-Integrated-4.0" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE" type="System.Web.Handlers.TransferRequestHandler" preCondition="integratedMode,runtimeVersionv4.0" /]

Yup, that makes sense. I think your previous IIS settings were incorrectly handling requests without extension (typical of REST API). I'll see if I can put a mention in the FAQ for anyone else that may come across this problem.

Thank you for sharing! :)

Yeah, I suspect I made an error when updating the Handler Mappings in IIS, it's probably ultimately easier and more reliable to just adjust the web.config file directly. Thanks for all your help!