Leonardo existed in an age where the key players were few and everyone of import knew each other. He didn't have to recite past accomplishments, because all the people he was writing letters to already knew what they were.
Had he been an unknown nobody, he'd have had to bring additional context to his appeal for employment just like everyone else.
Actually, he wrote this employment letter in 1482, before he made anything of serious artistic importance. He may have been somewhat known in the artistic community, but he was by no means very well known.
I mean, here in this age, nobody's going to know who you are as an artist before you make something important. But in the 1400s, just being an artist is a huge enough accomplishment that people are going to know who you are. Also too, what's important to us now is not the same thing as what would have been important to 1400s upper crust society.
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.
It may have been smaller, no doubt, but this statement is not correct:
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.)
Had he been an unknown nobody, he'd have had to bring additional context to his appeal for employment just like everyone else.