If that is the cause, shouldn't all actors in future dramas set in the Elizabethan era, the Stuart period or the English civil war should speak with a General Southern American Accent from now onwards?
Why does everyone in something set in Ancient Rome have British accents instead of Italian accents? Yes, I know Classical Latin didn't sound Italian, but it sounded far more Italian than English. Some things are just conventions by now.
I should add two clarifications to your post.
First, and I assume this is fairly obvious, but I should say it anyway. By old English, you mean Elizabethan English (which is an earlier modern English). It was after the great vowel shift. Before that point, everything was pronounced (and pronounced differently). The word Knight literally pronounced the k, n, g, and t and the i was like the i in the word Nick. That language would sound alien to us.
Second, more to the point. Elizabethan English was closer to a southern accent, but the southern accent has changed as well. The article there doesn't take a firm position on what accent would have been spoken. The recent discussion about Richard III demonstrates some British accents that predate the colonial period that are close to what is there today. I've heard in the past that the Tidewater accent is the closest, but the article you linked to seems to dismiss the idea.
I think what we do know is this:
The accent wasn't an RP accent. If a similar accent survives today, it'll probably be a lower class accent.
It was a non-rhotic accent. Most (if not all) accents in England don't pronounce their R's. Similarly, with the exception of Philadelphia and Baltimore (Mid-Atlantic accent), the American accents on the east coast are the same. The Mid-Atlantic accent owes its use of the R to a significant German and Irish population. The rest of America followed suit (either because they were settled from this group or out of a conscious decision to distinguish themselves), but the accents derived from English accents don't pronounce R's.