They say Ajax is back.

Ajax isn’t back.

Not back to the 1971-73 run of three straight European Cups. And not back to the 1995-97 stretch when they won the Champions League, lost the final on penalties the next year and went out in the semifinals the year after that.

Not back, even, to 2003, when one of Europe’s most decorated clubs came within a whisker of the Champions League semifinals again, with a team of Zlatan Ibrahimovic, Wesley Sneijder, Rafa van der Vaart, Christian Chivu, Maxwell, Steven Pienaar, Nigel de Jong and other assorted talent.

[ Follow FC Yahoo on social media: Twitter | Facebook | Tumblr ]

Ajax has reached the semifinals of the Europa League this season. It missed out on the Champions League by falling to Rostov in the qualifiers. But it did scrape past Legia Warsaw and FC Copenhagen in the knockout rounds of Europe’s second-tier tournament.

On Thursday, Ajax managed to give away the 2-0 lead it had taken in a scintillating performance in the home game against Schalke a week earlier. But in extra time, following a shaky and anxious Ajax performance, it went behind. Somewhat miraculously, though, the 10-man visitors then got two goals of their own to make it a 3-2 loss that was good enough to advance to the semifinals on a 4-3 aggregate score.

This is the deepest European run from the Amsterdammers in almost a decade and a half. But to pretend that it lives up to the heights the club once resided at, before the economics of the game made it impossible for a club from a lesser league to compete consistently, sells short the club’s legacy.

Sure, in spite of departures from Jasper Cillessen, Arek Milik, Mike van der Hoorn, Viktor Fischer and the talented Riechedly Bazoer and Anwar El Ghazi, Ajax has dazzled the way it hasn’t in years under new manager Peter Bosz this season.

Academy products Davy Klaassen, Matthijs de Ligt, Donnie van de Beek, Justin Kluivert and Joel Veltman have impressed on the continental stage. Defender Davinson Sanchez and attacking midfielder Hakim Ziyech were real finds on the transfer market. Brazilian winger David Neres, just a few months at the club, could prove a revelation.

But this will be no vintage Ajax team. Because there is no path back to the European elite.

How talented this team might turn out to be, we’ll never know exactly.

Because it doesn’t work that way anymore. Ajax players will be spotted by bigger clubs abroad and they’ll realize that they can make multiples more money there – in places where the television and sponsorship money dwarfs that in the Netherlands. Such players might resist such a move for a while in the interest of their development, but eventually, they’ll go. Before their primes truly begin.

You can hardly begrudge players for seeking out all the market will bear for their gifts.

But it leaves Ajax stuck in a perpetual rebuilding project doomed to never become the finished deal. Its best academy players and most promising young signings leave when they come good. And then the club is relegated back to its academy and a transfer market where its name doesn’t carry the weight it once did. At best, it can sign underrated young talent. Then the cycle begins anew.

On Thursday, Schalke stormed out of the gate in a wide-open game, with Leon Goretzka volleying just wide and Max Meyer curling his finish off the post before the clock read three minutes. At the other end, Nick Viergever managed a limp bicycle kick and Bertrand Traore had a goal disallowed for offside.

But Schalke would be the more assertive and decisive team in regulation. In the 53rd minute, Goretzka initiated an attack and got onto the end of it as well, putting his finish underneath goalkeeper Andre Onana, who was hardly without blame.

Three minutes later, Sead Kolasinac whipped in the cross on another marauding run of the left, which was converted by Guido Burgstaller.

Captain Benedikt Howedes came close with a header from a Meyer free kick, and 10 minutes before extra time, Veltman’s idiotic block on Nabil Benteleb got him sent off with a second yellow.

In the dying seconds of regulation, Daniel Caligiuri’s cross was blocked into the path of Howedes by Onana, but the Cameroonian managed to claw it back into his possession before the defender could get to it.

And then, in the 101st header, Caligiuri seemed to have decided it for Schalke as he managed to sneak onto a high ball between two defenders and nod it home.

In the 111th minute, however, Ajax snuck through to the next round yet again. Viergever, whose low shot had been parried well by goalkeeper Ralf Fahrmann seconds earlier, brought down a cross and managed to redirect a clearance back into the net from close range with his outstretched boot. The away goal would see Ajax through.

In the 120th minute, Amin Younes even got an insurance goal, putting the thing well out of reach. He cut inside from Ziyech’s pass and beat Fahrmann to his near post.

Ajax’s glorious run in the Europa League continues. And for the sport’s romantics, it is a lovely sight to behold. But no matter where this campaign strands, Ajax won’t reclaim a permanent perch among Europe’s best clubs.

Soccer’s prohibitive financial gap won’t allow it.

It’s all a mirage. A flashback.

Ajax isn’t back. Ajax probably never will be back.

Leander Schaerlaeckens is a soccer columnist for Yahoo Sports. Follow him on Twitter @LeanderAlphabet.