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Pattern matching callbacks: why can't an INPUT with MATCH feed an OUTPUT that is fixed (no MATCH or ALL)

It seems that the typical use case for pattern-matching callbacks is to associate a dynamic input and output together via MATCH (or ALL). One could create multiple instances of this INPUT/OUTPUT combination. However, I have a use case where I have many INPUT buttons that all feed one text div output. I would have a very simple callback definition if I could do this:

    Output("output_div_id", "children"),
    Input({'type':"generic_btn", "index":MATCH}, 'n_clicks') # this would match any of several buttons.

It seems this is not allowed as the Output needs to have MATCH in it as well. Why is this? Am I missing something? Without this, my Input list needs to contain all the button inputs which is not very scalable.