If I was to guess, I'd say the likes of Murtough and Woodward went back to the Glazers and asked for more money. I don't think the Glazers would have specifically told the club "we need to sign Antony and here's the money to do it". I doubt they even knew who Antony was.
I mean, Ten Hag wanted to sign Antony, but sanctioning the outlay came from above...isn't that how every club works? No manager signs off on money. And it's not on finance guys to say if they're worth it. Ten Hag and everyone 100% knew what we were paying for him, and that it was beyond the budget allocated.