Hmm. I'm probably missing something obvious, but I don't see why that is a problem? I can even see some upsides if that happens (reduced block variance, higher block density during peak times...)
The case I'd be worried about is if it became more profitable to re-mine the best-block to get those txfees, than try move the chain forward.
(Though just turning off alone is absolutely a problem too-- because it means that the chain temporarily stops gaining security, and it's advantage vs a lower hashrate attacker which simply doesn't stop is reduced).
Reduced block variance is bad too: the convergence of the bitcoin consensus algorithm comes purely from variance, the fact that when there is a chain split even if the hashrates are equal on each side eventually one side will get lucky relative to the other and get decisively ahead from the perspective of every node even given propagation delays. Lower variance, higher reorg length. If lower variance were acceptable due to fast propagation and/or tolerance of reorgs, then it would be much better to simply reduce the interblock interval than have mining randomly come in synchronized bursts as soon as enough fees show up.
Higher block density I agree could be useful, depending on if the system is more limited by peak traffic compared to overall volume. Both sorts of limits exist in the system, e.g. strength against selfish-mining like attacks care about maximum block propagation delays in the face of adversarially constructed blocks ... while catchup/IBD costs care about overall volume. To the extent that the latter is limiting it would be good to have a daily swing in capacity.