access to funds applications that are in the Blackrock and associated companies and banks
Yes in principle, but also not exactly.
(This post is somewhat off topic, sorry for the derail and feel free to skip!)
Blackrock and similar companies are asset managers. They run the investments for pension funds, ISAs, life insurance funds and similar. Mostly they invest that money in bonds and shares.
Depending on when in the bond/share's lifecycle it is bought, the money might go to the company, or it might go to whoever is currently holding that bond/share. In the latter case, even though the company is not receiving the money it is still indirectly affected because demand for its existing shares/bonds affects its ability to raise money by selling new shares/bonds, and if its share price gets too low it is at risk of a hostile takeover or activist investors getting a large interest.
In the past large investment companies didn't care that much about ethical concerns unless it affected investment returns but in the last couple of decades there's been an increase in "ESG" investing both a pull from investors saying they care about this and will put their money with managers who consider ESG factors, and a push from the risk of headlines like "Blackrock is a major shareholder in SmashingSealsForFunInc"
So it is correct that if Blackrock or other large investors decide that whether a company puts gender ahead of sex is something they consider significant when investing that is something companies will respond to, but it's not because Blackrock or whoever have funding that companies apply to get in the same way they might apply for government grants or whatever. It's being driven by the attractiveness of the company to investors.
This is not just a US thing, in fact ESG is now probably more significant for inesvtors outside the US because in the States there's now a push back, with ESG being associated with the Left/Democrats and anti-ESG with the Right/Republicans. Like pretty much anything in US politics that can be simplified into a soundbite it's become a tribal issue, so Trump, obviously, is going to go with a loud, crass and un-nuanced anti-ESG position, and US firmed are going to re-oritentate themselves to that. (Hence also the recent changes at FB).
Appreciate this is a bit of a derail, but I think it's important to understand the mechanics that are incentivising companies.