The next question would be, if the keychain just hands out the keys to the application to be used? How would it ever identity a program as being eligible to receive a certain key? An attacker could just ask the keychain for the key.

A user is commonly able to read the process memory of any process running under his account. An attacker could extract the keys from process memory. And so on and so forth.

So you are very likely again where you were before: An attacker being able to execute code in the context of a user can steal his data. The only thing that varies is how much work it is.

In theory Apple may have exposed a security processor akin to a smart card with which Signal could generate public/private key pairs where the private key resides in the hardware and cannot be extracted. It does not appear the MacOS key chain works that way.