But if those in power at the Camp Nou, where Barca suffered their fourth straight defeat to Madrid for the first time since the Beatles were breaking through in the 1960s, decide to take this performance as an indicator of Koeman's credibility -- not his legacy -- as someone in the right frame of mind, making the right decisions and showing his players the correct way forward, you would understand if they believe there are better options.
One of the symbols of this game came from Luka Modric, Madrid's 36-year-old midfield maestro. It was not just the fact that he devoured Frenkie De Jong and had Gavi for afters, nor even that he consistently made it look like the visiting side had 12 men.
The iconic nature of his afternoon, in which he mixed technical skill, streetwise nous, intensity, use of space and a sheer winner's mentality, was seeing Modric with his arms outstretched in front of him in admonition to his teammates.
Variously the arms-akimbo image said: "Why didn't you finish Barca off?" "Why didn't you make the right pass?" "Why are we still living on the edge in a game that should be put to bed?" Indeed, one the arms-out beseechments might have asked: "Why aren't there three or four of me instead of just one?"
It was that kind of afternoon. Tantric football. The delayed gratification came in a flurry of foolishness and failure, with Madrid finally getting the two-goal lead their football had merited -- even after that, Barca scored through Sergio Aguero -- but Los Blancos were worthy winners.
Carlo Ancelotti's men had a better mentality, made better decisions, showed a better work rate, executed better passing and there was always a better chance they would score.
In a word: better.
But back to Koeman. Under him, Barcelona keep getting caught by sucker punches. Not just against Madrid, but that is what counts here. Across their last three Clasico defeats, there have been a plethora of occasions when opposing runners from midfield or the back line "appear" up in the spaces that Barcelona fear being penetrated, yet neither the coach nor the players seem to learn.
Last time at Camp Nou -- exactly one year ago -- it was Modric and Fede Valverde popping up in scoring areas because they timed their runs beautifully and Barcelona either failed to spot their ideas, or did not bother reacting.
Ditto the first goal the Catalans conceded at Madrid's Alfredo Di Stefano stadium in April. Just as he did to score the second goal on Sunday, Lucas rampaged down the right from a full-back position, enjoying time and space to set up Karim Benzema.
Make no mistake, Madrid are a sensationally good counter-attack side. The word 'transition' is en vogue right now and it's not a bad one. Whether under Ancelotti or Zinedine Zidane, they are very good at going from a defending "all-hands-on-deck" situation to one where either you create danger, or actually score.
Madrid play with verve and make quick decisions that offset the knowledge that their midfielders are not the quickest sprinters; instead, what the likes of Modric, Kroos and Casemiro are asked to do -- and to perfection -- is think and pass quickly.
That was enough to undo Barcelona on Sunday: Speed of thought and speed of action met speed of fleet foot, and what resulted was Madrid's fourth straight win over their biggest rivals for the first time in almost 60 years.
One of Koeman's main flaws, and this should be sufficiently condemnatory on its own, is that the whole of his team is still less than the sum of the parts. Never mind this being Barcelona's first Clasico without Lionel Messi on the club's books for nearly a quarter of a century, it is the quality and quantity of effort without him that counts.
Rivals of quality now fully expect to punch into holes that Barcelona leave thanks to a high defensive line and the fact they consistently lose the ball in situations that might be far from their goal, but which adept transition teams see as a "thank you very much for your naive generosity" moment.
Meanwhile, there is no question that Memphis Depay, very definitely a Koeman favourite, has an ego that outstrips his decision-making. Yes, he's talented. Yes, the Catalan crowd yearns achingly for someone to love and adore now that Messi has been sold down the river by the club board. And, yes, he can play.
But none of that excuses the fact he is sloppy in possession and makes on-the-ball decisions fuelled by his wish to be centre stage and adored. When his run or control puts him into an opportunity area, Memphis looks immediately for a defender to square up, to dribble past.
A good coach -- or one who was paying proper attention -- would be seeing the positives and eliminating the negatives. However, just as was the case recently when Memphis gifted the ball away and Atletico capitalized to score, so the team from the other half of Spain's capital took the same route on Sunday.
With the score 0-0, Memphis moved with the speed of thought, and limb, you might expect from a Sunday league footballer, he lost possession to David Alaba. Three passes later, following involvement from Vinicius Jr. and Rodrygo, Alaba was re-supplied the ball and, having run free of any defenders, produced a world-class finish.
The decisive second goal came from a yet another goal-producing burst upfield -- the fifth such concession in Koeman's three Clasico defeats as manager, for those counting -- with Marco Asensio showing clean heels to his pursuers and then, after a Marc-Andre ter Stegen save, one of those kind of "frozen in horror" moments.
There was only one possible area the ball could rebound and still present menace. Lucas Vazquez knew it, the crowd knew it and everyone in the press box knew it, but Eric Garcia only really reacted once he felt the "whoooosh" of Lucas sprinting in front of him. Aguero's subsequent consolation goal was just that.
Whatever counsel Barcelona president Joan Laporta received from sporting advisor Jordi Cruyff about giving Koeman more time, whatever financial penalties the club's cash-strapped board are trying to avoid, there is an unavoidable truth to confront.
Some of Barcelona's promising players are not being properly taught.
Koeman has become more adept at fighting for his continued employment than at correcting individual and collective flaws, which means there is little individual improvement. Witness Frenkie De Jong, who looks stunned about what is going on around him and who should, really, be carrying a greater weight of responsibility and leadership.
Madrid deserved their win and it will be an irritant, long after Modric laid his angry arms to rest, that it was only a one-goal margin. As for Barcelona, they are no more than treading water under Koeman.
If that is sufficient for them this season then, fine. It's their call. But Xavi Hernandez has won trophy after trophy with Al Sadd and has made it clear he is ready to return home immediately.
It is one thing to miss a blindingly obvious decision on the pitch in the heat of El Clasico. It is quite another to be myopic about a choice that stares you in the face, week after week.