Ah, yes. The Premier League. Come in. Take a seat. Let’s have a look at these figures then. First up: £1bn spent on transfers in the calendar year. And against that, as of matchday two in the Champions League, a record of played six, lost five against the rest of Europe.

Not that Arsenal and Chelsea, both of whom were restrained in the transfer window, can be singled out on the Premier League’s huge-collective-spend-versus-Uefa-coefficient-down-the-plughole scale. On a night when the champions and the team with most points in the calendar year were both well beaten, it was instead hard not to reflect on the sheer oddity of a league where the art of team building appears to have been lost in a blizzard of acquisitiveness, unexpected stasis and frenetically muscular island nation football. And where somehow for all the millions spent, the Premier League continues to offer up to Europe’s elite club competition such obvious, startling weaknesses.

Mocked by Germany, fleeced in France, this seems to be the role currently of the world’s most outrageously gaudy league. There is a famous Mitchell and Webb sketch where a pair of haughty young SS soldiers look down at their menacing uniforms, the skulls on their badges, and wonder, all things considered, if they may actually be the bad guys after all.

Swaggering in through the door, jewellery clanking, solid gold hat in place, the Premier League appears to be here right now to perform a similar function, the dopey old bad guys here to make other teams feel better, and to provide moments of hubris and underdog joy such as that experienced by Olympiakos on Tuesday night.

The final whistle at the Emirates brought an agreeable kind of bedlam as players ran across to the travelling support and club officials embraced on the pitch, this huge, stunning glass-and-steel arena reduced, with painful poignance, to a rattly, high-spec shell. At the very least we can console ourselves with one thing. The Europa League, could, if England’s finest can pick up a few points somewhere, be about to become the best, most exciting Europa League in the world.

By the end of a slightly frantic 3-2 home defeat there was almost something comforting in the sight of such well-worn, familiar, top-down failings in an Arsenal team who are now probably heading out at the earliest stage.

Two weeks and two eminently winnable games into their Champions League campaign, Arsenal are now running on the reserve tank in Group F after this home defeat, which mirrored in so many ways their last Champions League home defeat, to Monaco.

Before kick-off the steeply banked sides of the Emirates were abuzz with the usual mild sense of expectation, lifted by the boisterousness of the away support in their seething corner at the Clock end. From the start, Arsenal’s lack of thrust was noticeable. This can at times look like a lovely collection of parts that seems content too often to sit back and admire distantly its supple movements, its advanced electronic parts.

Mainly, though, this was an Arsenal team that seemed to think it simply had to turn up to win this game, a peculiar air of unearned arrogance that, as against Monaco, quickly turned to anxiety. It was fed by the team selection. Resting players in Zagreb in the first match in this group had left Arsenal horribly open in midfield. Resting Petr Cech here made absolutely no sense at all. It was a game-changer, too, as Cech, Arsenal’s sole summer signing, the man who would supposedly lift this team though his sheer presence, sat on the bench watching politely as his understudy, David Ospina, made a mistake that helped the game slip away at a vital stage.

Kostas Fortounis swung in the corner that put the visitors 2-1 up but Ospina was the chief mover here, palming Fortounis’s inswinging set-piece down and over his own goalline, a moment that it would be a dishonour to associate with Sunday league or park football.

Beyond that this was a strong Arsenal team, one that should have been good enough to win. It is question that seems to recur with baffling frequency, not least at home in this competition where they have now lost six of their past 13 matches. Why did Arsenal look so anxious, so tentative, so lacking in drive? One answer is simply that they lack drive, are tentative and easily become anxious.

Arsène Wenger defends David Ospina after Arsenal’s loss to Olympiakos. Guardian

Their only real point of thrust was in the interplay between Alexis Sánchez and Theo Walcott, who both made and scored one. Otherwise there simply seemed to be a lack of leadership on the field, embodied by the sublime, wonderfully talented Mesut Özil, a genuinely A-grade player who seems increasingly concerned with overcrafting every pass when what this team really needed was some drive and purpose and muscle from its classiest player. And beyond that some appalling defensive organisation, with vast tracts of space in front of the defence that helped game but limited opponents score three times.

Arsenal did belatedly begin to play with some constructive fury but as ever they were vulnerable on the counterattack, a weakness Francis Coquelin is often left to combat single-handedly, skittering about covering space like a man trying to fix a broken mains pipe and simultaneously replaster the scullery. Again Wenger took Coquelin off with his team behind. Again this left them soggier than ever in the middle. By the end here they were playing with an experimental triple strike force of Per Mertesacker, Joel Campbell and Theo Walcott, a moment of low comedy at the end of another bleak night for Europe’s most bafflingly ineffective league.