After much trial and error, and eventually guidance from a kind soul on StackExchange, I was finally able to troubleshoot the problem. The fix was to use ExportForm to deploy it as a CloudCDF. I initially avoided this option because I thought it would require a CDF player installation for users. Here is the final working code:
CloudDeploy[
FormFunction[{"Databin ID:" -> <|
"Help" -> "Enter the ID of a public Databin",
"Interpreter" -> "String"|>, },
ExportForm[Dataset[Databin[#"Databin ID:"]],"CloudCDF"] &], Permissions -> "Public"]
Still not sure why that option has to be set, but it is great to now know the fix.