Apple could simply reimplement APN and iCloud Files and etc over their own Private Relay connections, so that any carrier who blocks Private Relay ends up blocking all push notifications to all Apple devices. They have not yet done so, but if they do, either all US carriers will have to collude to block it — which could reasonably be considered anti-competitive behavior and would attract immediate governmental review — or they'd all have to allow Private Relay, as otherwise they'd bleed millions of customers due to being hostile to iPhone.

It seems safe to assume that Apple doesn't like user APN traffic being tracked any more than the rest of their traffic, and so I would imagine this is essentially the roadmap for all Apple Services provided to their users over the next few years.

It's also possible the Beta is just to see what happens when they make the traffic visible and possible to block, and having seen the carriers tending towards "block anything that inhibits tracking", they might simply target Private Relay 2.0 as "HTTP3 over 443/udp to http3.apple.com" and have that be an N-way reflector point for all customer-provided Apple services, including Private Relay and the rest as different channels over that multipath HTTP3 layer. They chose to implement Private Relay in a way that could be detected, and are measuring how neutrality-hating carriers respond. Assuming it's their final answer is not a bet I'd take.