So it's not uncommon for customer facing subcontractors to be given the gear of and wear the uniform of the lead company. In addition this makes even more sense for temps - who originally were suppose to just be the dudes and dudettes who filled in when someone got sick or had jury duty. And of course in these cases the person is typically still an employee - just of the temp/subcontracting company.
But what happens is that these drivers are clearly not temps (they stick around Amazon too long) and for a true subcontractor, the managers of that subcontractor would have a lot more say over the conditions of work for their employees (i.e. being able to mandate vacation/time-off for individual employees, exceptions to some of the stricter rules (like no car singing???)) as well as working for multiple companies, instead of just Amazon alone.
In this case Amazon was indeed a joint employer - they had too much control over the employee. If the union tried to negotiate with the subcontractor on behalf of the employee, they'd get told "we can't do that because of Amazon" but then Amazon would refuse to come to the table. So either the union had to give up, or even if the subcontractor diligently came to a reasonable agreement with the union - Amazon would abruptly cancel the contract and force the subcontractor into receivership or something. Then Amazon could wash its hands clean of the matter and start fresh by looking for a new subcontractor.
Perhaps an employee might still have some rights here - like filing against Amazon for being a party to a breach of contract or something - but I could easily imagine this getting signed away into binding arbitration and the employee promptly losing.
Fortunately, this ruling under the new joint employer rule puts an end to that kind of nonsense. Now Amazon HAS to participate or be punished like any ordinary employer - it can no longer hide behind some fly-by-night delivery service partner company.
The nice thing about the NLRB is that, like the Equal Employment Opportunity Commission, they have the ability to sue directly as a party (rather than having the employee being the plantiff) which bypasses any arbitration agreements (since the NLRB obviously never signed one). (Source: https://www.wardandsmith.com/articles/the-nlrb-on-what-employers-get-wrong )