I think it all depends on your location, where you are going, and why you think you will be Buggin, in the first place.
Most situations, I'm staying home.
If it's bad enough, then EVERYONE will hit the road. Gridlock.
Trailbike or mountainbike would get you there faster.
OK, so let's say you avoid the gridlock, or wait until everyone has run out of gas.....where is it you are going? To the mountains, down fire roads, totally off pavement? Then YES, 4x4 will be helpful.
Are you driving to Aunt Jane's who lives on paved roads? Then maybe not.
---------
Let me give you an actual scenario, which I think will resemble the chaos that will ensue as a panic forces everyone to Bug Out.
Due to my work, I am considered "essential" and MUST always be able to get to work (lucky me). Therefor I have always had 4 wheel drive vehicles.
We had a snow storm, I was on my way to work. There are hills and valleys, it is a suburban area. Cars were stuck in the bottomlands, unable to get enough traction to get up the hills. *Remember, if one car gets stuck, all are stuck behind them.
My 4x4 Toyota truck had a cap on the back. I had shovels, a couple bags of sand, and then threw in some cinder blocks and tool boxes to add weight to the rear. That was my normal procedure for snowstorms. I typically deflate my tires to give greater bite on the road surface.
Upon waiting, and watching the sheeple, I saw them try again and again to make it up the hill, only to be hoplessly stuck, slide into the curb, basically blocking the road. I pulled up over the curb, onto the grass/sidewalk, and motored right up the hill, past all the numbnuts, and made my way back onto the roadway. Before the armchair quarterbacks jump all over this and say I could have run over someone, give me the credit that I checked first, went slowly, and a kept a watchful eye for any pedestrians. There were Zero people on foot, as it was near blizzard conditions. I was in 4x4 Low range which is basically crawling.
OK, so, now, let's tie this in with a Bug Out scenario. People will be in a panic. Cars will wreck, blocking the road, run out of fuel, blocking the road.
We should plan on a real mess. If roads are totally blocked no one will be gonig anywhere. That's how I really see it.
However, should you be running along and making progress, and come upon wrecked vehicles, or any type of traffic blockage, I would really like the ability to pull off the road, and get around the whole mess. That is where I think 10 minutes of having 4 wheel drive could turn a hopeless situation into a problem solved.
If it's a natural disaster, you may have large trees down, blocking the road, or telephone poles, you will need a Chainsaw, a towstrap and a decent vehicle to pull the debris out of your path. The pulling capability of a 4x4 is much greater than 2x4, especially if you add a wet road surface.
I am NOT saying it's mandatory, or that you will not survive without it.
I am saying that it does have it's virtues. My 4x4 vehicles have allowed me to go thru flooded creeks due to the higher clearance, through 3 foot deep snowstorms, and have gotten me so much farther back into the wilderness I've been able ot enjoy palces I would have never hiked to.
I still say that the "Bug Out" option is the last thing on your list, and is a pretty desperate move. Put a lot of thought into it.