Well, breaking with protocol (if that is what is involved) would open up the possibility of US action that might inconvenience Ireland in different ways. For example, the many US charter flights carrying service personnel, and the occasional US military flights using Shannon, might just go somewhere else.
How you would identify other "US Government" flights I am not sure. Shannon and Dublin airports together see thousands of flights by US-registered corporate aircraft every year. Almost any of them could theoretically be used for US Government business - especially those owned by air charter/taxi companies. The essence of having an executive jet is that it does not as a rule advertise its ownership and those travelling by such means (e.g. senior business figures) would probably not welcome instrusive inspections of their personal aircraft. In such circumstances they too might prefer a stop at Prestwick or Gander instead of an Irish airport.
Much of the media debate tends to present the issue in a one-sided way, without much consideration as to what the implications of a policy of this sort might be. I am not saying that inspections cannot or should not be carried out, but I would suggest that people think a bit more about the practicalities involved.