It’s only high school level productions that make it all generic Tudor.
There are plenty of filmed versions of Shakespeare that play it straight. And Shakespeare generally is more familiar to the English-speaking public than the Odyssey.
Before we get Baz Lurhmann's Romeo + Juliet, you need to get Franco Zeffirelli's Romeo & Juliet (hopefully with better and clearer consent and no underage nudity).
We don't have that straight reading of the Odyssey yet, other than the obscure TV movie mentioned on this thread, which was the first time I heard of it.
With O, Brother, Where Art Thou, the Coens were relying on Gen X and older generations reading the Odyssey in school, but the classics have been deemphasized in American classrooms over the last 30 years, in favor of diversifying the canon and adding in more parsing of non-fiction texts and being able to spot misinformation and manipulation. (All good things!) My oldest, who took multiple AP English classes, read only two works of Shakespeare in high school (Julius Caesar and King Lear) and read some Aristophanes, but no Homer.
I think for a large part of the audience -- Gen Alpha, Gen Z and many Millennials -- the Odyssey will be a new story to them.