Is there something wrong with this? Email, fax, phones, telegrams, snail mail, etc. Everything was the new weird thing at one point in time.
Tweets have the side effect of being public. Sometimes that becomes the main effect. Emails and the rest of your examples don't have that property.
Transparency!
You've clearly never had somebody forward your email on you. All digital communication is effectively public by virtue of fast, easy copying.
There is a difference between public nature of tweet where anyone can go and read it vs public nature of email where you would need to forward it to everyone (or one of them would have to tweet it which is back to what GP said).
Tweets are limited in content, unlike other forms of communication their perminance is at the leisure of one American tech company.
Their visibility is as well.
You mean like gmail?
Gmail is just one company's implementation of an application layer sitting on top of SMTP. You are free to use any other company's implementation or even host your own and you will still be able to communicate with all other existing SMTP services. Not at all like tweets.