What do you consider "proper tune?" Because, I can assure you that I owned at least two M14/M1a hybrids that I spent a small fortune on in modifications that would shoot the pants off ANY factory NM or SM model M1a/M14 build. And, they still never shot as tight as my two AR10 multigun rifles at ANY distance out to 1,000m. You should pay a visit to many of the latter day competitions. 10-ring rifles are many and varied these days. What you see fewer and fewer of is the M14/M1a. It's a matter of obsolescence.
I think that there are very few people around anymore who can really fine tune the M14. My government match rifle, issued to me more or less at random never failed me in match competition. It had an SAK barrel, uitized gas system a accraglass bedding and NM sights but that was about it.
But it held all 34 shots into a 5 inch ring at 600 meters - so I find it hard to believe that any of the current rifles are even close to that gilt-edge performance.
Obsolete? Not really.