[DoktorEgo]

I don't see this mentioned anywhere here, but there's a website I came across years ago that allows you to do the same, but you just scan the sheet instead of mailing.

https://www.calligraphr.com/en/

Albeit it's not entirely free (pricewise or otherwise). I still think your second idea is better, where a learning algorithm "captures" handwriting and generates a font or more text with the same style. I can only imagine the implications in authenticity and forgery, though.