Really, survival of the fittest is a meaningless tautology. It was never supposed to mean the strongest or smartest survive. The point is that those that survive are those who are/were those fitted to survive the best. The fact that they have survived proves that they are also the ones that were best fitted (to their individual circumstances) to survive. Anyone that survives can claim to be a winner of survival of the fittest. The people that get aid from the government, for instance, can claim to be the fittest, because they survived, and they acted in a way that in their conditions helped them to survive. Additionally, we have no idea what genes will be good in the future.
Ignoring the many flaws with the conception of the survival of the fittest, we have two points. One, any society deviates from the 'natural' survival of the fittest - 'those who can't stand on their own two feet' is often something determined by society, with or without government policies. You think a person born into a rich family who only survives because their family pays for expensive treatment noone else could afford embodies the spirit of survival of the fittest better than a person who survives because they are born into a country with free healthcare?
Secondly, is that the most important thing for the survival of man in the future, and the condition of our future society, is the type of society we have. Generally speaking people aren't meaningfully too different, and nurture has a lot of power. Specifically for this conversation, a society that has a 'survival of the fittest' mentality will not progress as much as a society that works with cooperation, and where we help each other, so even if we 'improved' the gene pool, we would weaken the environment it is born into, which would do more harm.