I suppose, then, it's for those who don't want to be tracked, and not a "very stupid idea" per se.
But then again, if you don't trust the remote to know who you are, then why do you have an identity with them? I mean, the remote service is SUPPOSED to know who you are. That's kinda the point.
Paid services necessarily require a higher level of trust (since you are handing them money) than random internet services. So we are off-topic from ssh keys and identity.
If you don't want someone knowing your personal payment details (CC #, billing address), then pay in cash and use services don't deliver things to your home. And if you can't, then just don't use a service.
But that's living in way too much paranoia for most of us.
But then again, if you don't trust the remote to know who you are, then why do you have an identity with them? I mean, the remote service is SUPPOSED to know who you are. That's kinda the point.