The described issue sounds as a known one that we have just fixed - maxFileCount doesn't change, but files deleted via API are still counted towards total. Basically, if a file is deleted by clicking the Remove button, the Browse button is visible. But if the file is deleted using the API, then that file is counted for the MaxFileInputsCount condition. It is set to 1 in your case, so the Browse button disappears after deleting the first uploaded file.
The issue has been fixed and the fix will be included in the Q3 2015 release.
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