Yasopp was famous long before he became a Pirate also. It was because of that fame, that warranted Shanks seeking him out to ask him to join his crew. Zoro was famous in East Blue as well, which is how Luffy learned about him. Mihawk being famous before the great Pirate age, doesn't mean he was yet World renowned.
If you have all these people inspired at Roger's execution to go out and become Pirates to make a name for themselves, that is prime people for Mihawk to seek out and challenge, so why wouldn't he want to take to the sea as a Pirate to look for those who have also taken to sea as Pirates due to Roger's execution, which allows him to move closer to his goal?
And yes, you're correct, part of Mihawks job as a Shichibukai is to be a deterrent to Pirates. The Shichbukai, who are Government sanctioned Pirates, that are allowed to be in that position in exchange for a cut of their profits they make from Piracy. When we first learn about this, Mihawk is the one used as a reference.
Are you trying to say that the Pirate who is used to lay the foundation for what the Shichibukai is, is not actually a Pirate? Why would Oda explain this to the readers like that, and make Mihawk the basis for this standard if he wasn't actually a Pirate? That makes no sense from a narrative perspective. It'd be like if Oda used Whitebeard for the first time to set a standard for the Yonko (which he did), who are the Four great Pirates of the Sea, and it turns out Whitebeard isn't a Pirate.