I don't think it's fair to say the ASA has no power. It's pretty good at regulating other mediums (ask Irn Bru, they've had enough ads banned by them!). They just seem very behind in how they control social media advertising.
I found this online:
"Last year, the Committee of Advertising Practice (CAP) introduced the Poster Pre-Vetting Procedure. This means that if
the ASA upholds a complaint against a poster ad on the grounds of taste, decency or social responsibility, the advertiser must pre-vet poster for the next two years."
I know our main issue is influencers declaring an advertisement AT ALL (before we even get stuck into taste, decency etc. (Hello teepee adverts!)) but similar to the above, maybe if an influencer continues to be reported to the ASA, they could be put on some kind of posting ban or review system (by Instagram) for their future posts. They would certainly take notice then. This would help weed out a lot of the shady ex reality stars and their slimming pills etc too.
Or maybe the ASA should be forgotten about for social media all together and there should be a whole new regulator that encompasses all social media posts, regardless of country of origin. Something the big hitters like Facebook, Twitter, Instagram et al would need to run. For them, surely the draw is that they're losing potential revenue by posters using their platforms to advertise free of charge anyway. So they could both regulate and generate more income by enforcing something.