"Another big problem with Bitcoin at scale is that you can't count it without access to the keys that let you spend it." I'm not an expert, but I'm pretty sure

"Another big problem with Bitcoin at scale is that you can't count it without access to the keys that let you spend it." I'm not an expert, but I'm pretty sure one could use the private key, without revealing it, to sign a message, for example "%date I, %owner, own the bitcoin in $account". Then anyone else could check the signature against the public key on the blockchain. Would that not be enough?