At Arsenal, Arteta has the final say over transfers. If Arsenal signed Anthony for £82m, it would be because Arteta wanted us to - that would be on him. When the terms for Mudryk got stupid, the club allowed Chelsea to beat us to the player. When the price for Rice got stupid, the club ponied up the cash. It’s very clear why - Arteta viewed Rice as key part of his plans.
I completely understand that that no modern Manager is solely responsible for any transfer. The games hasn’t worked like that for decades. But you didn’t actually answer my question - does ETH have final say over transfers? I’m asking because I don’t know. The answer may be that you (and by “you” I mean United fans in general) don’t know either, but it’s a genuine question.
I don't think anyone knows for sure, but I doubt any individual has "final" say on transfers, and I imagine that to be true at Arsenal too, even if it's come out that Arteta "has" that power. It's very likely to be a mutual decision between the manager, sporting director, and chief exec. (or whoever else controls the budget).
As for your examples, I don't think they're comparable, and I think you've speculated a fair bit about Arteta's role in them too.
Antony, by a fair few reports, was scouted while Solskjaer was still here and valued at around £25-£30 million by the club. Ten Hag came in and wanted a left-footed right-winger as a key part of his plans. The only players we had registered at the club fitting that description and being remotely suited to senior football were unptoven and out on loan at Sunderland, or suspended indefinitely, pending the outcome of a police investigation.
This caught the club off-guard as they hadn't anticipated the need to identify targets that weren't left-sided centre-backs, defensive midfielders (or indeed, Frenkie de Jong and just Frenkie de Jong) and allegedly a striker of some description. Reports have said we walked away early in the transfer window because Ajax quoted £50 million at a minimum, and we didn't think he was worth that. We ended up panic-buying him for much more, right at the end of the window, because we'd failed to identify anyone else who could come in that summer, had been absolutely humilated in our opening two matches, and by that time, Ajax had already sold other first-team players, so put the price up.
It seems we also had a few extra million to burn after getting Casemiro for a bit less than we were prepared to pay for de Jong (although this is speculation on my part), and given we were two games into the season, pulled the trigger on Antony, lest we remain light a "key" player. Obviously he's been shit, so all round it was a terrible deal as he's not come remotely close to being a key player and cost us a fecking fortune.
As for Mudryk and Rice, well one was an unproven kid who'd looked alright in the Ukrainian league, and one was a Premier League-proven midfielder, guaranteed starter for England, and (I believe) captain of his club. There's also your obvious need for a player in Rice's mould, while Mudryk would have been just another attacker. It still represented a risk at that price, but it makes far more sense for the club to authorise a big fee for Rice than it would have for Mudryk, and I'd be very surprised if Arteta was the one who single-handedly pulled the plug on Mudryk when the asking price kept increasing (especially once Chelsea got involved).
Using the hypothetical of Arsenal signing Antony, I still think it'd be different to how we came about signing him, because (I'm assuming) he'd have at least been identified as a potential top target for that position by your scouts, and therefore, potentially worth the outlay. For us, he was the player who'd most recently played in that role for our new manager, and the club hadn't found any alternatives so spent about £50 million more on him than we'd actually valued him at ourselves.