I've been a CTO at companies in various bits of the Anglosphere, and signed contracts in all of the bits (and some other places).
I'm also married to an Anglosphere lawyer (which gives me lots more exposure than I might have guessed to the idiosyncrasies of the profession).
In my experience, I very much agree with the headline (and abstract) here, but would emphasise that the US is the worst for archaic language, creating a worse barrier to entry. Most other countries in the Anglosphere have been through a "plain english" language push in both contract & legislation, while the US has stuck to "this language is proven through precedent" more stubbornly than anywhere else. This seems to be particularly prevalent in IP law, although that might be my tech bias showing.
I would very much argue against those (even in this thread), that say that the lay-person isn't the intended audience for lots of contracts, particularly for T&Cs where they are often written by a contracts lawyer, for a contracts lawyer or judge, but should ABSOLUTELY be written for the lay-person to understand.
Anyway, I think the legal profession can & should do better.