Is there a circulating $E$ field when a single straight wire cuts a magnetic field?

I am a little confused as to the difference between the Lorentz force and Faraday’s induced emf, specifically in the case of a straight wire moving through a uniform B field. I know the answer is that the Lorentz force applies and we can take the velocity of the charges to be that of the moving wire cutting through the B field so we get a force at right angles along the wire and therefore a current. This force is due to the moving charges generating a B field which then interacts with the uniform B field = force. And the force is always along the wire in this setup. However, the wire is experiencing a change in flux, if it is part of a complete circuit, so we should also get an induced E field in the wire. And this field is a circulating field.

I’m confused as I think we should get both a Lorentz force and an induced emf but I don’t think that is the case? When I’ve studied coils of wire we’ve only talked about induced emfs and not the Lorentz force as if the latter does not apply in that case. Can anyone help me understand what I am missing here?