Can a flight instructor whose instrument rating isn't valid legally offer instrument instruction?
A few issues here
Instrument currency - Per 61.56(c), you must be Instrument current to serve as PIC aboard an IFR flight or one below VFR weather minimums.
For a pilot receiving instruction to meet the aeronautical experience requirements for an instrument rating (61.65), the flight instructor is considered PIC for all dual training flights conducted under IFR or weather below VFR minimums (1.1). Therefor for all dual IFR training flights, the instructor must be instrument current to serve as PIC
Now an certified instrument instructor may administer training to meet the aeronautical experience requirements for an instrument rating. But this not need be done in IFR. An instructor can provide this training in a sim, FTD, or on a dual instruction flight conducted under VFR. In these circumstances, the CFII need not be instrument current to administer the instruction and the student can log that time toward an instrument rating.
So flor local VFR training flight with the student under the hood, this is fine. But if the flight is say filed as IFR, then the CFII must be instrument current to conduct the training.