Lets examine your "gigantism" reasoning a bit deeper.
First, lets assume its correct, and North America's fauna during the Pleistocene had an overall tendency to be larger than anywhere else. Presently there are no fossil evidences for any apes (other than humans, of course), large or small in North America (not even in Asia near the land bridge). So, the argument -if its core supposition is correct- is just a speculation.
Now, lets check the core of the reasoning- North America's fauna during the Pleistocene had an overall tendency to be larger than anywhere else. Well, it does not seem to be correct. Yes, there were species that were larger than today's, but the same was valid for other parts of the world not to mention that many species were shared with other places. Mammoths, for example, also lived in Europe and Asia; mastodons also lived in South and Central America. Giant terrestrial sloths lived in North, South and Central Americas and the same is valid for the giant terrestrial predatorial birds. Panthera atrox, the extinct North American lion (whose habitat extended southwards to Peru, BTW), the european cave lion (Panthera leo fossilis) and the European and Asiatic lions provide another example of a widespread genus. Europe and Asia had the Woolly Rhinoceros. To this, we could add bisons and elks, for example.
The specific example you cited, the saber-toothed cats (Smilodon genus) also shows that the argument is flawed. Smilodons were spreaded across the Americas. The largest species (S. populator) was from South America (3m, ~400kg); North American species (S. fatalis) some 30cm smaller. The North American lion could be larger but were of a lighther build (I found estimates varying between 2.5 and 3,5m, but remember that Smilodons had short tails). European cave lions were about the same size.
As a sidenote, some people say that given the bulk composition of the Pleistocene fauna, Africa is the last place in the world where the megafauna (or somehting roughly like it) still lives.
At last but not least, you could argue that the Pleistocene was a time of giants (the name megafauna has a reason, after all). But there's a drawback- setting the standards for "normal-sized". How can one tell, for example, if those animals actually were giants or if the current fauna is mostly composed of dwarves? Large animals quite often appear in the fossil register, after all.
I could go on and talk about other problems with the reasoning. The bottomline is that its core presuptions have flaws (North America's Pleistocene animals were not larger than those from somewhere else and there are no fossil records of apes -other than humans- in North America), and it fails to provide a backing to bigfeet as real animals.
A word of caution- there's no need to "explain" the size of bigfeet. What's needed are reliable evidence to back its existence. So far, none was obtained, as far as I know. Thus, the "not real" option becomes the dominant position.
.....and is a sign of fear...pure and simple.
