It’s true that trying to state complicated necessary and sufficient conditions will inevitably involve a ton of essential complexity due to the nature of the subject matter, but there may still be room for improvement by eliminating additional accidental complexity introduced by e.g. cumbersome syntax. I think by "legalese" the authors probably have in mind only the accidental complexity introduced by the distinctive, convoluted syntax of legal language.

For instance, here [1] is a random paragraph I found in a contract that I think is pretty good example of "legalese", and here [2] is my attempt to rewrite it for readability. All the essential complexity remains, but I think (hope!) much of the accidental complexity has been removed. :)

[1] 3.3.4 Date of Issuance. Each person in whose name any book entry position or certificate for shares of Common Stock is issued shall for all purposes be deemed to have become the holder of record of such shares on the date on which the Warrant, or book entry position representing such Warrant, was surrendered and payment of the Warrant Price was made, irrespective of the date of delivery of such certificate, except that, if the date of such surrender and payment is a date when the stock transfer books of the Company or book entry system of the Warrant Agent are closed, such person shall be deemed to have become the holder of such shares at the close of business on the next succeeding date on which the stock transfer books or book entry system are open.

[2] 3.3.4 Date of Issuance. To determine the record date for ownership of Common Stock shares (whether issued as a book entry or certificate), ask: Were the Company's stock transfer books and the Warrant Agent's book entry system open when the Warrant was surrendered and the Warrant Price was paid? If yes, the record date is that same date of surrender and payment. If no, the record date is the close of business on the next day when the books and systems are open.