When you use Put with a string as the location it just creates a regular file, not a CloudObject. What's happening is that viewing files with the cloud file manager is automatically generating the CloudObject metadata for those files. You'll want to change your code to either Put to a CloudObject explicitly, or preferably just use CloudPut:

obj = CloudDeploy[
  APIFunction[{}
  , CloudPut[$RequesterWolframID
    , "testFiles/" <> DateString["ISODateTime"] <> ".txt"
    ] &
  ]
, "apiTest"
, Permissions -> "Public"
];

Also, there's no need to create the target directory when going directly to cloud objects, since it will be created automatically (as a CloudObject) if necessary.

PutAppend supports cloud objects, by the way. You can try that, it should be better performing than having to check for the object existing prior to every write.

But yes, a CloudPut of a tiny expression is probably the fastest way to create a cloud object. (You can include the IconRules -> {} option setting, the icon is generated asynchronously so it won't improve your latency if this is done infrequently, but if you really are hammering the system, this will reduce the overall load on the system and may help some.)