Shakespearean English is not that common in usage unless you are watching a Shakespearean play/movie. Medieval English is NOT very intelligible with modern English at all in fact, and is why we do not use it in media. Urianger is sort of Shakespearing when he speaks... if it were on steroids. However, the titles you listed are all in modern English. Downton Abbey takes place in the 1910s and the Queen just a few decades later. Game of Thrones, the Witcher, and Dragon Age is just the Queen's English. It may get tweaked in some cases, but its actually English still being spoken right now and cannot be considered an older form of English. But my point is, that ffxiv uses an English a few centuries older and that can sound overly flowery or bland which was what the op was complaining about.
Here's a little comparison for you. I'm using a scene from John Adams and they read an actual proclamation from King George III followed by the actors discussing it https://www.youtube.com/watch?v=5JTxVHQAp8w. Now compare this to a simple scene between Aymeric to Emmanellain https://youtu.be/hl9AWSWBp94?t=1929. The English is very similar from how sentences are constructed to vocab usage. Of course this is not consistent nor is it perfect in ffxiv, but that is due to it being difficult to emulate when you are a translater pumping out hundreds of pages every few months. Now let me compare this to a scene from Downton Abbey just so everyone can see how 1700s English is quite different from the Queen's English https://www.youtube.com/watch?v=MHu6bmcx1cg. The speech is very posh but its not the same English that is used in the John Adams video or Aymeric's.
Of course, ffxiv does not stay in 1700s English all the time. The translators tend to fail at consistency because characters like say, Lyse, tend to speak more modernly. But I feel the majority of the cast uses vocab and sentence structures like Aymeric.
Also, I think that if we used English like Downton Abbey or the Queen, the dev team would have accomplished a better translation because simply put, its easier. I am not advocating for the use of modern day US English at all. I think that the softer sounding British English is a better fit for the fantasy genre. But the issue is because ffxiv tries to use 1700s English it puts a slight language barrier for many who play this game. While you might be good at listening to this type of English due to different exposures, my roommate is an average joe who constantly asks me 'what does this word mean'. Hell, my girlfriend who is not a native born English speaker at first had a hard time with ffxiv's English. She is now a pro at it, but it takes some exposure. But because we are using an English that is three centuries old, it can make them sound bland. That is my point and ultimately the ops issue.