But only Uber (allegedly) had access to the private entitlement. So how did other apps do it, and why couldn't Uber?

That said, this is certainly a "special relationship" with big partners like Uber to get their apps working at Apple Watch launch in 2015[1].

That the access "wasn't removed already if it was no longer required", well... I think we can all understand legacy code.

[1] "It is likely that the Cupertino tech giant had to give this access after it gave app developers a four-month window to develop apps for its Watch before the unveiling of the product."