from what I saw the allegations that people are bringing to say that hawks was the one to take down the HPSC are: (this are taken down from a post that I saw here on tumblr btw)
they doing shady business with redestro (this has nothing to do with hawks), they forcing hawks to do a mission he didn't wanted (okay, they did this but hawks never showed any thoughts of rebellion against them) and hawks keeping his humanity and trying to save hawks until the end (except when he killed him of course), so the redestro clone disappeared in the moment jin died, and the time it took hawks to save twice from dabi's fire allowed that clone to stab the HPSC president (but hawks didn't plan any of this so he didn't have any intentional role into taking them down)
Oh, thanks for the heads up and letting me know where the question comes from! And as I read, let me deconstruct very quickly and briefly the entire argument which is based on nothing concrete.
I. The black market for Hero equipment run by Re-Destro, as you said, has nothing to do with Hawks. He has never been involved with the business in particular, and the only relevant thing which Hawks has done in regards to warn the heroes and the Commission of their plans is to distribute the manifesto of the PFL in a way to let them know what was happening (and even then, not all of his information has ever been complete and/or entirely reliable).
II. Forcing Hawks to take the double-agent mission and later keeping his ‘humanity’ to save Twice until the end, is first of all not even an argument and second it really does not hold up with the narrative we have during the last moments of Twice. I am not going to go very deep into Hawks’ relationship with the HPSC, which yes he defines as a Golden Cage, but Cage nonetheless, but at the same time he actually condones its actions because they are made in order to protect a greater good and rationalising these actions may be the only possible coping mechanisms Hawks came up with in order to reconcile his persona with the soldier (even though I don’t think it would be entirely correct that he is a soldier for the HPSC, rather a usable pin who might become unnecessary - even though this exactly fits the meaning of a soldier who lives in order to fulfil their orders) that the HPSC presented him as his persona. It also means that Hawks’ chance to escape the Commission is to actively fight like Lady Nagant did and (this is an argument I’ll use later to confute the president clone claim) actively go against them, but at the same time even if he kills a higher-up, will that bring him satisfaction and will that change the situation? The only thing to change would be the person in charge and Hawks’ status, which would go from hero to villain, and the Commission would still flourish with the other soldiers it created. Therefore, Hawks accepting the mission and keeping on his mission is just proof (I don’t mean to say that Hawks is a coward, because that’s not what it is) that Hawks has interesting which transcend his own will to free himself from the shackles he has been bound with. As for Twice, there was no humanity whatsoever shown when he manipulated someone he even dared to call a friend, and then stab him in the back.
III. ReDestro’s clone stabbing the HPSC President is something that likely the villains planned, and that again confirms the fact that is actually the villains’ actions which have made a difference so far in the current situation of society rather than a plan from Hawks (as I wrote here), who instead just is reaping what the villains had sown. Again, his stabbing the President would have changed very little if the had not been chaos all across society and that many members of the HPSC would have been undisposed and many heroes decided to retire, because it would have just taken one hour in order to substitute the president and let the status quo remain. The thing is, the HPSC is an organisation which has many layers, and one cog less does not determine its fall down. This is why it is highly unlikely that this precise action is what took down the HPSC - but it was the ensemble of thing that happened, which shook society to its core.
I hope that this argument has been contrasted properly, because I did not want to go in too deep on the thematics which tie the HPSC as a Sybil System of sorts (if you saw Psycho Pass you know what I am talking about), where Hawks, while a victim of it endorses the system nonetheless because it might actually be the means to an end (or accomplishment of the greater good).
Thank you for the question and thank you for reading!