You are right that the force on the electrons is negated by the Hall effect. The short classical explanation is that the Hall-effect electric field also acts on the ionic lattice that makes up the rest of the metal.
In terms of classical electromagnetism, the explanation would go as follows:
When a current is flowing in the metal, there are two parts: the free, delocalised electrons, and the fixed lattice of positive ions through which they move. The electrons are, on average, moving at the drift velocity, while the ions are not able to drift, so only the electrons experience a Lorentz force. This causes them to experience a magnetic force perpendicular to the current, creating a small excess of positive surface charge on one side of the wire (where there are fewer electrons than ions), and a small excess of negative charge on the other (where there are more electrons than ions). This creates an electric field that balances the magnetic forces on the moving electrons, stopping further buildup of charge - the <PERSON> effect.
The electric field does not only act on the electrons, however. 395
It also acts on the positive ions, and because they are positive, the force on them is opposite the direction of the force on the electrons, i.e. in the same direction as the original magnetic forces! So, since the electric field exactly balances the magnetic field, the net force on the wire as a whole is in the same direction and (since the ions have the exactly opposite charge to the electrons) magnitude as the magnetic forces on the electrons.
This subject is discussed in much more detail in this paper, which presents a classical quantitative model for finding the force on the wire from the microscopic forces and the relevant Hall electric field. However, to fully describe these kinds of microscopic interactions, quantum mechanics is necessary. According to the linked paper, more advanced condensed matter textbooks develop the quantum version using the Fermi gas or Bloch functions. The above picture - where the Hall effect is all that is relevant - works in a metal where only one conduction band is present, but in semiconductors without a Hall effect, collisions between charge carriers and the lattice may be more important. 531