In this context, open source/self-host means do it yourself, closed source means you trust someone else to do it for you.
In the long run open source always wins because of the community effort, customization, network effects and price.
Internet protocols are open, anyone can setup a website, host their own email server..., but most people don't do that, they rely on someone else to do it for them.
People still pay for Windows rather than use Linux because most people and companies have better things to do.
The main threat to American AI companies is not that consumers will self-host, but it's that new hosting companies will appear that will host open source models and offer them (cheaper) to consumers. It'll be like the web hosting market before the era of cloud computing.