Diodes in an op-amp feedback
When the op amp output is at (nominally) 10 volts, the current through the diode string will be, as you realized, very close to zero. Then the op amp will have effectively no effect, and your analysis is correct.

The answer sheet is wrong. It is basically assuming that the op amp loop is working so as to keep the inverting input at zero volts, so with no current through the feedback resistor the output is asserted to be zero.

However, with the diodes not conducting, there is no feedback current to keep the inverting input at zero, so the answer sheet's assumption is incorrect.