https://johnsonsdictionaryonline.com/blog/copyright/ suggests they don't want you to download it

Of course they don't, but perhaps they ought to allow it, or even to make it easy to do.

Most likely they don't feel comfortable enough with the quality of their extracted data. The original text is in the public domain though, so it's only a matter of time before other replicate this work and make their version more easily accessible to others. For example, Wikidata is a publicly downloadable, general-purpose database which provides a way to enter detailed, referenced information about "lexicographical" entities, including sense definitions and quotes. So anything that's in the Johnson could easily be posted there.