I'm going to take "caveman" as someone living a hunter-gatherer lifestyle in primitive conditions. I think such a person would have a very busy mind, but not the same way I do. The HG would be constantly aware of his/her surroundings, even when relaxing. Many rest times would still be productive: While sitting around the fire yarning, they're still building arrows, spinning cordage, or whatever.
Our minds, on the other hand, are pretty much always running, even when we're supposed to be sleeping, usually about something other than what is happening here and now.
I think a lot of this has to do with the nature of caveman stress verses modern stress, and it's one of the reasons I spend as much time in the wilderness as I can get away with. A crisis in the wilderness is usually sudden, extremely intense, and over relatively quickly. Say I'm out in the back country, hunting solo, and I fall and bust a leg. Storm's rolling in. Well, I either stabilize my leg, manage pain, find shelter, get a fire going, find some water...or I die of shock and exposure. Probably within a matter of a few hours. Either way, the immediate crisis is over for the moment, and I either get over it permanently, or I move on to the next challenge.
Or say my elk calls bring in a griz or a mountain lion. Huge adrenaline rush, intense stress...and depending on the disposition of the critter and the skill/luck of my reaction, I either run the critter off or it takes me out. Either way, my problems are over, at least for the moment.
I think those are the kind of stresses we're designed to deal with. Deal with it and get on with life, or fail to deal with it and don't. Difficult, but pretty straight-forward.
Modern stresses, on the other hand, tend to be low-key but prolonged over a period of weeks, months, or even years. Modern life includes few big adrenaline rushes followed by quick resolutions, but abounds in those back-burner worries that just go on and on and on: Struggles with bosses, chronic illnesses, debt, impossible schedules, so many options for pretty much every aspect of life that we can't possibly make informed decisions. Modern life is all about excessive complexity. I don't think we're designed to deal with that low-key but long-term stress. That's the stuff that keeps our minds running all the time, that keeps us from living in the moment and experiencing the beauty right in front of our noses. Or as John Lennon said, "Life is what happens while you're busy making other plans."
So, is the mind of a hunter-gatherer clogged with constant, conscious thought like most of ours? I doubt it. It's still busy, though. Just with other, more immediate stuff.
T