How about creating public getter and setter for this privatePropertyOfApplicationTag and bind to this property?
Some things that are easy in mxml are impossible in ActionScript and vice-versa.
No. MXML compiles down to ActionScript, so anything possible in MXML is possible in ActionScript. You're right that the reverse isn't true.
To answer your specific question, you can declare bindings to private variables in MXML, but this doesn't really establish bindings--it only evaluates the expression (and in fact, the compiler will issue a warning to that effect).