I think we're mostly in agreement, but somehow your post is worded like you're debating me
My point was that taking the time off is unlikely to affect your career in the long term.
Your example about letting people go when the decision makers don't know their day-to-day proves my point - those people clearly aren't super valuable/are easily replaceable, otherwise someone would know what they do day-to-day and raise a stink when they appeared on the cut list. I work at a (really) big company, and I have a say in which people get promoted or let go, I'm speaking from experience.