As TimLymington and others have pointed out, the old, archaic words have strictly-defined meanings that a "more common" word does not have.
I'd like to also point out that legal documents must be unambiguous to an extent not required of most writing. Suppose in a novel the author writes, "Bob dropped by the office where his wife was waiting with her boss. He kissed her briefly and they left." Who did Bob kiss, his wife or her boss? Technically the language is ambiguous, but I think we would normally assume that the writer meant that he kissed his wife. If a few paragraphs later we learn that Bob is having an affair with his wife's boss and his wife doesn't mind, and so he is kissing her boss, we might be annoyed at the writer for the momentary confusion, but it's no big deal. We figure it out and move on.
But suppose a legal contract says, "Acme Trucking Company will deliver the shipment to the Beta Company warehouse. They will pay the invoice from Nadir Machine parts within 30 days." Who is the "they" who is supposed to pay for the shipment, Acme or Beta? Any reasonable person would say that of course the trucking company isn't expected to pay for the merchandise, that's the receipient's responsibility. But the text is ambiguous, and a lawyer could argue that the contract means that the trucking company must pay for the goods delivered to Beta. I'm sure a smart lawyer could come up with some plausible-sounding reason why this was part of the deal and thus the intended meaning of the contract.
I once got burned when a legal document that I was a party to mentioned a certain $4000 in assets, and then later in the document said that I had the rights to this $4000, but failed to clearly connect the reference that it was mine to the earlier statement describing it. So the lawyer on the other side said that this was a different $4000 and that the ownership of the first $4000 was therefore unspecified, and so his client was entitled to half of it. We eventually settled the matter by me agreeing to give him ... I forget how much ... toward the legal bills that his client was unable or unwilling to pay, and thus the client got nothing from the negotiations and it all went to the lawyer. But that's another story.
Anyway, my point is, legal documents are verbose because they dare not risk leaving any case not covered, because a smart lawyer for the other side may try to exploit any such ambiguity. While for most documents you could say, "Oh, just use your common sense", or "If you don't understand, ask me and I'll explain it," you can't do this with legal documents.
Could they be less verbose and use less mysterious language than they do? Probably. But it's not the slam-dunk easy thing to do that you might imagine.