This is a thoughtful gesture, but there are at least two problems with it
First, a handwritten note isn't easy to respond to. With an e-mail, you can leave the message in your inbox until you have time to respond and then it's one click to start responding. With a hand-written letter the recipient would have to context-switch from reading mail to using a digital device and they'd have to transcribe your e-mail address. It's not much work, but it's still work that someone has to make time for.
Second, it's an unusual thing to do. It's important to communicate with people through normal, comfortable communication channels where the etiquette is known. Having someone handwrite a letter and look up your mailing address is unusual. Unusual behavior triggers people's suspicions. You weren't trying to scam anyone, but you should be aware that one of the tricks used in scams is to invest unusual amounts of attention and energy into someone. It can trigger a suspicion that you're really after something else.
Your second round of sending short e-mails had neither of these problems. Easy to reply to, nothing unusual about it. It's the way to go.
Solid advice, (but to others reading) don’t optimize for not irritating people.
Especially your professional network. You built those relationships because you wanted to use them in the future, and the future is now… so, tap them! If they don’t like that, no big deal, no sweat, move on.
If you’re irritating multiple people constantly, then yea you’re probably doing something wrong. But if it’s one person once in a blue moon, that happens, don’t over optimize for avoiding that. You can’t make everyone happy
i'd rephrase as "visible lack of effort is problematic" - anything above that passes the bar for me, and other factors become critical