Of course he isn't 100% responsible for it. The only thing he's responsible for is recommending Antony, and in a properly run club, the director of football and scouts look at their reports on Antony and go "not happening, these are our shortlisted targets for that position" (as Liverpool famously did when Klopp pushed for Brandt over Salah).
Even if you accept the repeated nonsense about Ten Hag requesting "full control of transfers" as a condition of accepting the job (which comes from a single, out of context quote), it's still on the club for accepting such a request. He simply should have been thanked for his time, and the next candidate for the managerial position brought in for talks.
After that, it's on the club for not having any alternatives lined up when it was clear that Ajax wanted silly money for Antony (apparently we balked at a quote of £50 million earlier in the window).
And after that it's on the club for authorising £80 million for him after the season started, because we'd failed to find anyone else for the position.
His eye for talent, given Antony's obvious lack of suitability for the Premier League can certainly be questioned, as can his apparent belief that he can still come good, but it's still got to come within the context of the above.
That's what's been reported. Everything I've read has said that Ten Hag arrived and (unlike basically every other professional club around these days) we didn't have what they call a "shadow squad" of targets for every position, should we find ourselves needing players at short notice. The club knew he wanted a midfielder in the mould of Frenkie de Jong (which they seemed to take as "Frenkie de Jong" and is how we ended up rushing through Casemiro when the penny finally dropped that de Jong wasn't coming) and a left-sided centre back (Martinez was on the shortlist), but were blindsided by his request for a right-winger as they (presumably) thought Sancho would fill that role.
Aside from the fact that it left us unprepared to find an alternative to Antony (we apparently decided £50 million was too much near the beginning of the window), it also meant we'd have been left in the shit with any other sudden departures. Be that a player picking up a long-term or career ending injury, your high-profile striker deciding to sit down with Piers Morgan and slag off the manager and the club mid-season, or (in an incident pre-dating Ten Hag) a young talent being recorded (allegedly) threatening to rape his girlfriend.
When you think about the farcical "500 scouted right-backs" thing that somehow pumped out £50 million for Wan-Bissaka as the best option, it's obvious that the recruitment team was a complete mess, despite apparently containing 160 scouts.
A director of football's job isn't just to do what the manager says. If that's what happened, then it's still not on Ten Hag.
He didn't approve spending any money on Antony (or any player). The team negotiating the transfers (likely led by Arnold, and previously Woodward) approve transfer spending.
I don't even want the man to stay in the job (at least not past the remainder of the season), but there's enough stuff to criticise him for without pinning the results of our shite structure on him.