Why would that require running your own cell network? The Verizons and T-Mobiles of the world would be happy to stick another cell phone in their roster. The only barrier is getting people to buy it.
Edit: Actually there are plenty of third party app stores for Android already - including several from popular phone manufacturers[1]. The main barrier is probably getting developers to put their software on these stores and getting consumers to buy from them.
Consider the Parler case study. There the app was removed from the app stores but also Amazon stopped servicing the application. You could imagine a scenario where if the app stores had decided to not remove the app, then the underlying carriers could've been leveraged to disable the application.
To truly escape the walled gardens of the cell phone world, it would seem you need to control the entire stack from carrier up to device and application.
