To get back to Gutcher's original question: have Mackay's conclusions been refuted in more recent times?

I don't know if any study of his conclusions has been published, but it doesn't seem too difficult to find serious problems with his approach and the conclusions it leads him to:
1) He seems to have decided from the beginning what he is going to find, and then - no surprise - he claims to have found it. In the first place, he doesn't give reasons for thinking that the refrains are anything other than what they seem to be: melodious nonsense. And he doesn't give any reason why, if they do mean something, they should be druidic chants and not something else out of all the many possible things people have sung over the centuries.
2)His choice of Gaelic as the language of the supposed chants flies in the face of all that is known (and was known in his day too) about the differences between Celtic languages and their geographical distribution. As has already been pointed out, almost certainly the language spoken by British druids would have been more like an old form of Welsh,and the Celtic languages spoken in other parts of Europe were related but different again.
3)Having decided on Gaelic, he uses only the modern Gaelic of his own time. (I haven't seen any indication in his book that he was familiar with older forms of Gaelic or had much sense of the historical development of the language, but I may have missed something.) This leads him, for example, to assume that the "dh" in "luadh" was always silent, and so derive the "loo" in "fal lero lero loo" from it, but it almost certainly wouldn't have been silent 2,000 years earlier -- as with English words like "knight", the silent consonants represent sounds that were still there when the spelling of the words was established.
4)Not only are his supposed Druid chants based on modern Gaelic, they are just strings of separate words as if taken straight from the dictionary, with almost no sign of any grammatical connection between them – hardly any prepositions, no changes in the form of the words to indicate how they relate to each other. It is only in his English "translations" that the words are actually made to make sense.
5) The Gaelic words he chooses may have some similarity to the sounds in the refrains (and any difference can conveniently be explained as the result of "corruption"), but that alone is hardly enough to prove a connection. As he has started out with the idea that they are Druidic chants, he picks words that he imagines might fit in a religious context (apart from the reference to the oak tree to explain "down down derry down", I don't see much that is specific to the Druids about them). But they could easily be matched to other Gaelic words. Already he is uncertain whether "derry" is from "darach" or "doire", but why couldn't it equally be from "dara" = second, or "dearbh" = certain, or "deiradh" = end…? He derives "fal" from "failte", but why not "faile" = smell, or simply "fal" = turf? And so on.

So, sorry, he doesn't convince. I reckon he would have done better to have stuck to journalism.