Yours is a fair question. Arebelspy is surely our best authority on the forum on how well it works out post-FIRE, and I'm betting he'll have some good insight for you.
My experience pre-FIRE is that by far the most time I spend is in the actual acquisition of the properties, and in the immediate aftermath of that when prepping for and placing the first tenants. The actual buying of a property is a few dozen small investments of your time (filling out loan applications, digging up and submitting documents, arranging for an inspection, reading the inspection report, negotiating concessions, etc.). If units are empty or there's a tenant that needs to be moved out, there may be some rehab necessary too. Truthfully, I think I have a tendency to micromanage rehabs a little too much. During a rehab I probably text or email anyone I'm actively working with every couple of days to make sure they're on track. My completely honest assessment of how much I spend on a property during purchase and rehab is about 4-5 hours a month, though I could probably reduce that if I set my mind to it. On properties that are already up and running, the time commitment goes way, way down. I haven't answered one call or email (aside from checking my monthly statements when those come in) on my fully occupied properties in at least a few months.
If it helps to set your mind at ease, even in the case you cited where I fired my PM, there wasn't too much time involved. I sent one email to my PM 30 days before I wanted to terminate our contract (CCing the new PM), then one more on the day of to remind them that my new PM would be by that day to pick up keys, rent rolls, leases, etc. There were a few annoying maintenance issues that the old PM had allowed to go unaddressed that I had to approve over the following few weeks, but nothing too terrible.
I guess this is all a long way of saying that it's possible to spend very little time once things are set up and you trust your team. It's also possible to spend a fair amount of time if you make a poor decision on a team member (as I did). My complaining on my blog aside, it didn't eat up too much of my time. I just spent more money than I would otherwise have had to, and consider it part of my learning curve. By the time we RE, every property will have been running for at least a year or so-- that should give us the time we need to shake out any kinks.
I think if I were in your shoes and wanted to go off-grid for a few months, I would either have my dad be my backup contact, or allow the PM to make the decisions, making clear that I didn't want cosmetic items addressed until I returned.