But the reality of it is he did and based on that reality, it is far more likely that he had lots of kids than just 1 or 2.
Yes, it is
more likely that he had more male children than was necessary to become Y-Adam, but that does not mean that such a thing can simply be assumed; "probably" is not "certainly". All we can state with certainty is that he had at least two male children; all else is blind speculation, and no sound basis for anything at all.
Based on evidence the time period between this guy and the migration out of africa was a mere 10,000 years or so, a blink of an eye in evolutionary terms. That alone makes it more likely that he had to have a head start, basically lots of offspring. I would say between his time and the migration, it'd be 700 generations, and probably less. There just wasn't enough time for thousands of generations to pass.
Even seven hundred generations is still enough time to render the actions of any one individual candidate for Y-Adam largely irrelevant beyond a certain point. As I said, we have no particular reason to assume that any given lottery winner bought five tickets rather than just one.
On the second point, he probably had these kids over an extended period of time and back then, boys started to hunt at a very young age. Larger groups during that time are actually shown to be more able to survive, so having more kids wasn't a bad thing.
But you are not merely suggesting that he was supporting many children in a row, but at once, a far more difficult proposition, especially given that Y-Adam seems to be, in your conception, a self-interested womaniser with little very limited interest in the off-spring which his gallivanting actually produces.
Also there were probably other males in the group that didn't reproduce nearly as much as him but hunted also.
And why would they share the proceeds of their hunting and gathering? Gatherers in simple societies, especially, tend to consumer their food as they gather (they gather/ed far more food than they hunt/ed), and they're unlikely to put up with some revolting little sprog who keeps following them around and eating their finds just because his dad is the local Big Man. Aside from anything else, it would be in direct contradiction of the Social Darwinism you deem to be natural; if anything, they'd strangle the little bugger and dump his body in the undergrowth, lower the competition for their own off-spring.
Also, as sons of the highest ranking male, they most likely ate before the other kids(or at least their father got the best pickings and handed it down to his preferred kids).
...Thereby blowing your Social Darwinism out of the water by reflecting the irrational hereditary privilege leant by social hierarchies.
The actual trace in lineages stops with him. The evidence actually suggests that all males in that group actually had his specific Y-Chromosome. Each lineage's Y-Chromosome is unique. They might have decendents in the group, but those guys obviously didn't survive very long. The thing is that from Australia to America, no trace of other Y-Chromosome lineages could be found, so there a decent possibility it was only his descendents.
Even that is less than certain. There could have been several such lineages, all sharing an even earlier Y-Adam. All we know is that each of these lines of direct male descent terminated at some point, and, while this was presumably relatively early, it doesn't suggest that our contemporary Y-Adam's Y-chromosome was ubiquitous at so early a point.