To what extent would Bitcoin be private without address reuse? Let's say Bitcoin had these rules from the start:

An address with balance cannot more receive money
An address must use all its money in one transaction

This would prevent address re-use. To what extent would this make Bitcoin anonymous?
This would not help, and would actually make things worse. Bitcoin transactions are public, the input and output addresses are known. This can therefore be traced. Simply tracking the transaction from an initial input (say, buying currency on an exchange that complies with Anti Money Laundering (AML) laws) through each of its single outputs to a destination address would de-anonymize any destination. Since addresses can't be re-used in this scheme mixer services (money laundering addresses) become impossible, so anonymity is impossible for anyone who uses an exchange compliant with AML laws.