I think the deepest motivation of correctly setting your online/offline state is to allow the blockchain to adjust internal parameters related to the cryptographic sortition.
The sortition has some optimal expected values that want to achieve for safety, and your total population size should be tuned correctly to make the math work within those expectations. In particular, if a specific set of tokens are sampled to participate in the committee, we try to assume that all of them will be online for voting.
This doesn’t mean that the online stake should be 100% accurate since some thresholds will still make all this work. In the worst case, this would affect liveness but not safety.
I think that’s why setting your online/offline status correctly is essential, even if your case is offline stake. You’re helping the blockchain calculate better the sortition (in both cases).
This is my interpretation.