You start a classic movie and after a few annoying previews a lion's head shows up, surrounded with three Latin words: Ars gratia artis. What does it mean? The short answer is that it should mean 'Art for art's sake', thus being a translation of a 19th century French motto "L'art pour l'art". There are some concerns about the validity of this translation. Priamarily, Classical Latin does not have a single word that would correspond to our idea of 'art'. However, the Latin of personal and family mottos is a very mixed substance. It is not uncommon to have a motto that displays Medieval influences or bends the meanings of Latin words in some way. It's not ideal, but it happens. I would argue that a plural form of ars (artes) would have served better in the MGM logo, but I think it's passable as it is.
The words of this logo are also reminiscent of "Ars longa, vita brevis"('Art is lasting, life is short') - a translation of a phrase from Hyppocrates, certainly not from Classical times. While the Greek term 'techne' does not really mean the same as 'art' in English, the entire phrase (both in Greek and in Latin) gives us a clue that 'techne' and 'ars' can be aplied to something of perrenial value. Now, what do we have left from the Ancient world? What has maintained its value? Quite naturally, that would be art (in our sense). So, the use of 'ars' by MGM is not that outrageous.
By the way, Howard Dietz, a 19-year old recent dropout from Columbia's School of Journalism, who was asked to come up with a logo for the new movie studio, ended up having a career within MGM.