Endwalker was "meh" (for reasons other than technical issues)

[Jandor]

The technology existing is fine, despite the qualms I'll have about it's physical representation being so underwhelming. That, however, is aesthetic taste that is 100% subjective. The biggest issue that rung to me was the fact that the aether that supposedly was used to power the teleporter was completely drained just previous in the same scene in order to shatter the keys that kept Zodiark in check. Failing to break the last one with the very blatant remark that the aether had been depleted. Considering that the trek between earth and moon, regardless of technological efficiency, is certainly no small feat, I struggle to believe there was still enough for him and Zenos to be sent to it's surface. That's not even mentioning that we were sent up there as well. Whether I misunderstood the scene and mayhaps it uses the aether of the individual which would explain the WoL and Co. heading up there would justify that latter bit, but Zenos being Garlean and therefor unable to process Aether and Fandaniel stating that there was none, potentially implying the machine, himself, or both are out, made the whole thing make little sense.

Garleans have aether, they just can't manipulate it innately.

Zenos however, has been able to manipulate aether for a while, he found his innate inability to do so an annoying vulnerability in his youth and was one of the few people willing to sponsor Aulus's research into giving Garleans the ability to wield it.

Regardless though, Garleans are apparently able to teleport using external machinery just fine. It's mentioned in the Garlean base that they use what are essentially the same aetherytes that we do, they just require a machine to cast the teleport spell for them.

Further to all that, it's also established earlier in the story that it's possible to configure aetherytes to send people to other destinations, even if they lack magic and have never attuned to the destination. One of the first things we do in this expansion is chuck Thancred in to one in order to get him to Thavnair. An ascian/telephoroi/allagan/whatever teleporter achieving the same thing is not unreasonable.

Edit: and to add, it's established back in 1.0, and later reconfirmed in Stormblood IIRC, that teleporting uses anima. That particular resources relationship with aether is pretty damn vague, but the machine expending all its aether trying to blow up the ward doesn't mean its teleporting ability would cease to function, as it doesn't necessarily require much, if any, aether to work.