Replace “teller” with “bank” because we are talking about legal ownership, not physical control.

They don’t have anything but a responsibility to care for it

While they absolutely “have a responsibility” to you, they also benefit from holding it, so your “anything but” rhetoric is incorrect. Brokers and banks alike earn money by lending the assets the have, despite their corresponding liabilities.

Do you think there isn’t a “clear record” without direct registration (from your other comment)

Correct. Legally, you have a “security entitlement”. Per UCC 8-503, the property interest you have a result of this entitlement is merely “a pro rata property interest in all interests in that financial asset held by the securities intermediary”, i.e. what your broker actually has, which is (a) opaque to you as a customer, and (b) is fundamentally difficult even for them to pin down - as it is composed primarily of their DTC account balance, ideally but they undoubtedly have many derivatives, transactions to settle (which can extend beyond 2 days because FTDs are common), shares lent out that are due to them, etc. So while the number of security entitlements in your account has a clear record, your property interest in the issuer does not have a clear record.