We don't let the phone company disconnect phone calls in which people complain about the phone company, or about China, or about the governor. We don't let FedEx decide that you can't ship an unauthorized biography of the FedEx CEO.
Common carrier status has never applied to news media. The closest we ever had was the fairness doctrine, which is gone and wasn't the same thing anyway. Fox News is clearly news media. Most of what we call "Google" is clearly not news media.
It never applied to websites either, so why do you want it to apply to websites but not "news media"?
Do you believe that Fox News should be allowed to delete comments from foxnews.com?
I don't think so. It's the same old stuff, merely with a technology upgrade. Be thankful, so the first amendment doesn't only protect quill pens and lead movable type.
The comments on foxnews.com are at the borderline, so it wouldn't be bad to flip a coin. A reasonable approach would be that deletion is restricted to crude insults and off-topic rants. It doesn't really matter though, because the foxnews.com comments sit on the border between being a publication and being a generic public communications medium. There is no point splitting hairs.
Verizon's phone network is a platform that is privately owned and financed by the owner.
I think you're failing a litmus test here, unless you really believe that Verizon should have the right to drop phone calls that are politically offensive to Verizon executives.
The same goes for FedEx's logistics/transportation network. It too is privately owned and financed by the owner. Imagine that it were run by an executive who decided that the terms of service would include disposal of democrat campaign materials. Send a box of Biden yard signs, and they go into a dumpster. The tracking number just disappears from the system.
It's probably hard to see that Google is very much like Verizon when you happen to like Google's bias.
So from that perspective, why should Google be treated differently from Fox News simply because Fox News has a higher standard for publication than Google?