No, for four reasons.
1) For the most part, EME does not encode more information than modern English.
If you find words like "afeared," "forsooth," or "sportive" to be more evocative than their modern English counterparts, it's probably because they take on a certain luster from their archaic nature in and of itself. If people still said "afeared" instead of "afraid," it wouldn't have that attraction, or the exclusive association with EME writers like Shakespeare. It's also to some extent a class marker; using an antique word shows that you've had access to education.
On the other hand, if we now always said "ocean," and "sea" had fallen out of use, Shakespeare's "sea of troubles" would sound different to our ears.
2) In those cases where there is more information encoded, English speakers got rid of that information for a reason.
For instance, English speakers started using "you" and "your" for the second person singular, replacing "thou" and "thy". If it was more important to English speakers to make that distinction, they would have kept it. They thought it was easier to have one word for both.
3) Having more information encoded in a single word isn't always a good thing.
English is as rich as it is because it has a large vocabulary. If you want to say "afeared," you have dozens of synonyms to choose from: frightened, spooked, creeped out, scared, terrified, et cetera, et cetera. Many obsolete EME words are obsolete because they encode too much. For example: "anon" can mean soon, or it can mean imminently, or it can mean now. It's generally more useful to use any of those three more specific words than the vaguer "anon."
4) Vocabulary, like grammar, is descriptive, not proscriptive.
This is the most important reason. Words aren't defined by grammarians; they are defined by the people who use them to communicate. No "movement" is going to change the way people speak, unless it's backed by a strong central government that also has charge of the educational apparatus of its speakers. See, for example, PRC character simplification and German Rechtschreibreform.
It's reasonable to say "People never say 'wot' anymore."
It's reasonable to say "I wish people still used the word 'wot'."
It's not reasonable to say, "Let's bring back the word 'wot'." Unless you enjoy fighting in the windmill-lists.