Well ... I think it makes sense they expect you to have an Irish passport if you travel as an Irish citizen
Even if you were born Irish I think no EU country would let you in just based on your Irish birth certificate for exemple - so I don't think hte fact of being naturalised makes a difference. The rule you quoted clearly says you need a passport or an ID card; and the country where you are travelling needs to give you the opportunity to obtain it if you have forgotten it or lost it.
If you travel with your other passport, you are traveling as a citizen of that other country and therefore need a visa.
If you are really in a rush, I am not sure about Ireland but I know some European countries can issue "laissez-passer" or emergency passports for emergency travel which are essencially a travel document valid for a short period within the EU in case you don't have a valid passport and need to travel urgently:
http://en.wikipedia.org/wiki/Travel_doc ... _passports
I saw that at the airport in Paris once. A French guy in front of me had lost his passport and had a temporary citizenship certificate which seemed to be valid EU wide (we were flying to Dublin). The Aer Lingus staff were not aware of this and had him wait to double check whether they could accept it as a travel docuement; and after 5 minutes they came back and said there was no issue and let him check-in.
Maybe you can research whether Ireland can do the same thing?