did paul just let the Judaizers and other false teachers just keep preaching on though?
Didn't he exhort us to expose their darkness by bringing the light of true gospel into their teachings?
Paul was an Apostle writing to the churches. That office no longer exists.
Yes. We are to expose false teaching and darkness. That is done through the preaching of the truth and confronting false teaching when it is encountered.
As far as playing the part of false-teaching police; we are not called to do that. We cannot patrol social media for every bit of falsehood. We cannot stop people in the workplace who are following the prince of the power of the air (Eph. 2:2). When given the opportunity to speak the truth in those venues we should take it, but we cannot spend all our time doing that.