Basically, it's a cover letter.
If your actual resume consists mostly or exclusively of theoretical future accomplishments, employers will be justified in wondering whether you've actually successfully done anything. If your cover letter consists mostly of rehashing what you've done in the past rather than what you can do in the specific role, employers will be justified in being sceptical about whether you've actually considered whether you're actually the candidate they need.
Had he been an unknown nobody, he'd have had to bring additional context to his appeal for employment just like everyone else.
The main point I'm trying to make is that 1400s Europe was a really small world. So different than the one we find ourselves in that, if you want to take anything at all from Leonardo's cover letter, adopting his language will probably get you much farther than adopting his actual technique, which was devised to fit the needs of the 1400s.
But in the 1400s, just being an artist is a huge enough accomplishment that people are going to know who you are.
Artists in da Vinci's day were more akin to craftsmen. Da Vinci himself was basically raised in an artist's workshop (by Andrea del Verrocchio) and worked alongside dozens of other students on their master's projects. This is in stark contrast to today's artists, who are basically expected to be the sole creative genius (barring people like Damien Hirst, of course, who is known to use assistants.)
The author seems to be annoyed by people sending him resumes instead of cover letters. Maybe he should learn what a resume is before complaining about the ones he receives.
A cover letter should address a specific vacancy and explain the relevancy of the candidate to this particular job at this company.