He literally didn't, being stronger swordsman and being stronger are 2 different things.
Pica used sword vs Sanji first, then he dropped his sword and focused on his Haki only, and thats fodder Pica trying that, whose Haki wasn't even highlighted like Vergo before.
Saint Shanks's Haki gets hyped almost every single time he appears, only a moron would deny Saint Shanks doesn't have better Haki than Mihawk, if he does the same thing that fodder Pica tried, Saint Shanks would smash Mihawk without needing sword skill. And for sure, Saint Shanks's Kamusari doesn't cut, and Saint Shanks wouldn't really need a sword to replicate same feats. And same Kamusari style user Roger, wasn't WGS despite being PK and equal of WB.
Also, both Alber and Sanji confirmed using sword doesn't make someone a swordsman too. Saint Shanks can be like Alber, no one says he couldn't.
Saint Shanks literally says he doesn't play by the ''rules''.