No-one knew whether viewers would accept a "serious drama" about power politics with a fantasy setting. This article shows how much of the early criticism and rejection of the show (particularly from mainstream critics) was essentially a knee-jerk reaction to the fantasy genre, and how the producers did famously try to pitch the show as "Sopranos in Middle Earth", emphasising the politics and drama over the fantasy setting.
I thought there was an interview where one of the producers said explicitly that toning down the fantasy styling where possible (including dialogue) was a deliberate part of making the show appeal beyond fantasy fans, but I can't find it now I look for it (please comment with a link if you have one!).
A couple of further observations:
Using modern language for normal dialogue helps us see what is non-modern from the characters' perspective
The show does preserve some medieval or fantasy-style language, but saves it for things that are mystical, legendary or traditional from the characters' point of view, or which reflect some feature of the fictional world. For example:
Show-only expressions like "Gods be good", "Seven hells!", "It is known" and "the iron price" stand out from normal dialogue and flesh out the religions and traditions of the show's world, helping keen viewers to see and understand cultural differences between characters.
Phrases like "The night is dark and full of terrors" contrast with normal dialogue, making them more obviously a non-literal traditional set phrase. You don't need to know anything about Melisandre's religion to know when she's reciting dogma, it's obvious in the phrasing. There was another great example of this in season 6 episode 1:
Sansa stumbles over the archaic (to our ears) wording used to accept a formal offer of service from Brienne, needing a reminder that she must unrealistically promise to share "meat and mead at her table". Even if we don't recognise the phrase from previous seasons, the non-modern language helps us understand that she's struggling to play the part of a noble lady and honour a formal tradition verbatim
Elaborate fantasy-esque character titles like "the sword of the morning" stand out from normal descriptions, and tell the viewer that the person or thing referred to has legendary status within the world of the show. When characters like Dany give themselves such titles, the contrast with normal speech tells us more about their aspirations and character.
Fantasy critters are described using outdated or unfamiliar words by characters who are skeptical or dismissive (e.g. "snarks and grumpkins"). Making it sound old-fashioned or make-believe to our ears helps us understand that this is how the characters see it, too.
Medieval-style etiquette is sometimes added to normal dialogue, usually when it tells us something about the medieval-style power or status dynamics. For example, "My Lord" and "m'lord", the politics around who calls who "your grace", various medieval ways of showing deference such as "if it please", etc. When characters drop these formalities and address each other in a normal, modern sounding way, that too tells us something about the relationship (e.g. when Marge was befriending Sansa and the language between them got progressively more modern and casual).
This wouldn't signify as much to the viewer if everything the characters said sounded equally old fashioned or fantasyish to our ears. Obligatory XKCD cartoon.
Understanding the characters' position in and perspective of their history is an important part of the depth of the show, and one the books can build through book-only tricks that don't work in a TV show like using internal dialogue for exposition, using quote marks around set phrases, and long asides.
Unfamiliar words are less jarring in a book
It's easier to get away with using non-standard language in a book because readers can look up a word they don't recognise, slow down while reading a difficult passage, or skip back to re-read something that didn't quite make sense the first time.
There are several paragraphs in the books that I struggled through and in some cases had to re-read after I'd figured out that, for example, a "destrier" was a type of horse, a "capon" a type of game meat and a "cog" a type of ship.
In a TV show that is already pushing the boundaries in terms of the number of characters, storylines and things to remember, anything that reduces viewer confusion will help.