[c_of_red]

'Hidden' units 'pop up' from time to time. Mostly because a change in suppression gives them a better searching roll, which they then make. Logic says that in order to make their new roll, there has to be a new roll in the first place. So one can deduce that the change in suppression has created a new search roll for units within LOS. So back in the day, it was normal to see if there was LOS to a certain hex. Sometimes in the process of right clicking to change facing and check the LOS, a 'hidden' unit would lose it's hidden status. So one deduces that it was the change in facing that generated the new search. That led to the 'click 360' scouting technique.
If the search has a 6% chance of success, doing it 6 times won't increase that 6%, but it will give you 5 more chances to make that 6%. Think of it as noticing something but not being sure what it is so you take a closer look.
So I think the real question is if that 'show visible hexes' button calls the search routine like changing facing for an infantry unit does. I think your answer was no.
I'm not sure about that because if there is no LOS then you cannot 'see' a unit, which makes the LOS critical to seeing a unit.