Blocking out, say, 2x2 hours each day for "problem management" (fixing root causes of recurrent issues) becomes easy in that situation, with no subterfuge needed. Alternatively, you can nominate part of the team to be either on-call or entirely dedicated to incidents, shielding everyone else who can be dedicated to strategic work full-time (obviously, you need to do regular rotations).
The team themselves often need help in adjusting to that (firefighting can make you feel like a hero so, although stressful, it can be addictive). Blocking out specific timeslots can be helpful from this point of view, too - incidents often seem a lot more manageable if they're not crashing in on top of you at unpredictable intervals.