I've taken mental health leave (not due to a PIP) and my productivity before and after was significantly different. It was great for my employer that I took it. I'm quite sure I would've eventually ended up with a PIP if I hadn't taken it sooner myself, and the best remedy on a PIP would have been to take mental health leave. Not as a strategy as such, but literally because it would have been the best solution (and I think the only one).
Mental health problems are tricky; they tend to creep up on us gradually, and often some form of external trigger is needed in order to prompt us to seek help. So it shouldn't be at all surprising that an employee in receipt of a PIP might take mental health leave as part of a genuine effort to improve their situation.
gp's cynical "counterfactual" suggests that they view PIPs as being purely a sham, intended to always result in dismissal rather than improved performance. Now, that might occasionally be true - but we should be blaming the abusive employer (who is likely acting outside the law) in that situation, not the employee.