Seems that we can’t both get what we want.
A potential solution is that you get your anonymous phone line but my phone provider simply refuses to let you call me with it.
Of course then we need to extend the same principle to data and to IP traffic originating from your device. If you don’t want to be traceable it seems reasonable that services should have the right to refuse to handle IP traffic you generate.
Would such a half-baked level of network access suit your needs?