Literally smalls claims / arbitration works.
Apple is happy to do this because even when the arbitrator decides they're in the wrong it'll 1) be much faster than court 2) be much less than a jury awards. There are pretty much 2 arbitrators in the US and so they can afford to lose a single client.
Its only uphill if you have an actually bad argument such as _just_ the statement "they broke our TOS.".
Additionally (I am not a lawyer) arbitration does not override any court or have any authority unless authorized by a court. If a party does not cooperate with the arbitration, the other party still has to go to court to convince a judge that the arbitration was good and get the judge to enter a judgement against the noncompliant party.
In terms of the arbitration itself, it is very hard in the US to get an arbitration award overturned. So, wouldn't bank on that.