Isn't this already happening? Well, not exactly with one public address, but with one master public key. You can create nearly unlimited addresses which aren't linked and there's neither interaction from the sender.
Anyone who has your extended public key can see all your transactions, anyone who has your silent public key sees literally nothing. That is the difference.
I still don't understand how silent payments improve anonymity. Doesn't Alice still have lots of outputs in different addresses? Didn't she have the same problem before?
Previously, Alice had one address with many outputs created by different senders, with silent payments Alice has many addresses, each of which with just one output (provided that it is an ideal implementation that prevents address reuse). In the former case, all outputs are linked, in the latter they are completely delinked.