In theory, are there algorithms where everybody can read a message but only one person can write it?
If you can decrypt it, it is the right sender and if you can't, someone must have altered it / is not trustworthy.


Then by definition what you're looking for is called a signature. You're asking for something which is a signature, but is not called a signature. Sorry, there's no such thing. A signature is called a signature. And verifying a signature is called verification, not decryption.

It's not that “the signature method is the way to go”. It's that the properties you want are that of a signature. Therefore, anything that does what you want is a signature.