The issue presented in the article is not with the verbose definitions within legal texts. Precision is, indeed, important when defining laws. The issue is, however, with center-embedding, which involves nested definitions, often with multiple layers, in the legal text, making it difficult to parse and which can even create ambiguity.
An example of center-embedding:
"The officer who the witness whom the prosecutor questioned identified failed to appear in court."
This sentence may describe an officer who failed to appear in court, the officer may have been identified by a witness, and the prosecutor may have questioned that witness. There are many ways that center-embedded sentences could be re-written. Their clauses may even be independent and therefore separable into individual sentences.
Are the embedded structures the product of an afterthought?
What happened? "The officer failed to appear in court"
Which officer? "The officer, who the witness identified, failed to appear in court."
Which witness? "The officer, who the witness, whom the prosecutor questioned, identified, failed to appear in court."
Are they the only way to link the definition to the noun in the sentence?
Surely not, especially given that many legal documents "declare their variables", so to speak, in a prepended "definitions" section.
Are they meant to make legal documents more difficult to parse and understand by laymen?
This is the most likely answer unless it's just a weird, old habit of the legal profession. Either way, we can do better.