Could it be that by paying in lawful money the Purchaser acquires substantial enough of a title to the pertinent property to make the Deed of Trust (i.e. assignment to bank) solid?

I have thought about this before. A contract is a contract with any consideration. Coin, no matter how worthless, or cotton paper and ink are enough. Only a party to the contract can question such a thing, otherwise it is tortious interference. The only thing left is presumption of a bad contract, but paying the asking price (no matter how trivial) of the previous owner gives one perfect equity and that is as good as allodial title. So, unless it is only for the false presumption, or court enforcement (silver and gold at least to give a lawful judgment), I can't see why 'lawful money' would be in there.