How to solve the "Set a password"-step problem in a really user-friendly way?

I'm an advocate of lesser cognitive load process, moreover when this process includes lots of forms and steps. I think we all are at some point.

The password step by itself if usually full of pain point because some users are not technical as we are (obviously) and when you ask them to use at least one special characters, and you don't aspect the "_" as a special one, well, it's kind of disturbing.

My point isn't in the "what should be a special character?", but more of a "why do we really care of what the user typed in this field?"

My questions are the following:

• Why would you ask for a precise pattern for a password, since you know it adds a lot of cognitive load for the users, as well as the need to handle error cases in terms of design and development?
• Is there, in your opinion, a better way to do it?

Below, my thoughts as a first answer. Feel free to vote up or complete by answering.