There's definitely a God...

I think if early man, living in his hostile environment had the savvy of today's average person, the sabre-tooths would have got fat and bored pretty quickly.

Not at all, the sabre-tooths would have acted as an environmental pressure and very rapidly left only the agile and aware amongst us, perhaps in as little as a few generations.

Plus - of course we could have never got to the stage where we were picked off by predators at will unless sabre-tooths were a sudden introduction into our environment (or us into theirs).