Until I saw Mike's link I had never seen "a real McCoy".
While the dictionaries read "the public", it is correct to say "a public".
While the dictionaries read "the real McCoy", it is incorrect to say "a real McCoy".
From time to time we remind members that there is no 'Academy' to decide what is correct or aceptable in English.
Modern dictionaries and grammars record what native speakers actually say and write, not what they should say and write. Some lexicographers and grammarians give advice, based on their findings, as to what is generally accepted as standard usage, but such advice is not binding on anybody. Writers of style guides and of publishers' 'rules', may prescribe certain usages and proscribe others but, once again, the general public is not obliged to pay heed to them.
People who respond in forums such as this often give very useful advice but, as you see, we sometimes don't agree. That does not necessarily mean that one of is wrong. It usually means that more than one suggestion can be acceptable to many native speakers. When it comes to how the indefinite and definte article are used, you may find that in a group of half a dozen native-speaking teachers and grammarians, no two will agree exactly on the 'best' article in every place in some texts.
Learners who speak languages for which there is an 'Academy' which passes judgement on any and every question about language usage may find this surprising, but you'll have to accept it. There is often no point in worrying about why this is so.