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The answer is an unequivocal yes. Quantum computers would definitely still be useful.

Quantum computers are not oracles for BQP, but rather devices which process quantum states, and can communicate using quantum states. Just as the ability to make non-deterministic queries is fundamentally more powerful than the ability to make purely deterministic queries, independent of the status of P vs NP (and indeed this is the root of the oracle separations), the ability to make quantum queries and to communicate using quantum states is fundamentally more powerful than the purely classical counterpart.

This leads to advantages in a wide range of applications

The ability to query oracles or external databases in superposition provides a provable separation between quantum computers and classical computers in terms of query complexity. There are a variety of communication tasks which see drastic reductions in communication cost which quantum communication is used. Quantum information processing allows for information theoretically secure protocols for a wider range of problems than are classically possible. Certainly QKD does not require a universal quantum computer to be implemented, but many protocols for other tasks do. Pre- and post-processing of large entangled quantum states allows you to violate the shot noise limit in metrology, resulting in more precise measurements.

Aside from the complexity arguments, there is another practical reason to want quantum computers. Much of the data processed on classical computers these days is derived from sensing the natural world (for example via the CCD in a digital camera). However, such measurements necessarily throw away some information about the system in order to render the measurement result as a classical bit string (for example collapsing spatial superpositions of photons), and it is not always clear which information will later be considered the most important when initially recording the data. It is, therefore, reasonable to believe that the ability to store and process quantum states directly, rather than collapsing them in some basis prior to processing will become increasingly desirable.