Ideas and correctness depend on each other. You usually start with an idea, and check if it is correct. If not, you adjust the idea until it becomes correct. Once you have a correct idea, you can go looking for more ideas based on this.

Formalisation and (formulating) ideas are not separate things, they are both mathematics. In particular, it is not that one should live in Lean, and the other one in blueprints.

Formalisation and verification are not simply certificates. For example, what language are you using for the formalisation? That influences how you can express your ideas formally. The more beautiful your language, the more the formal counter part can look like the original informal idea. This capability might actually be a way to define what it means for a language to be beautiful, together with simplicity.