"For relationships as distant as third cousin (2 x (1/2)8 = 1/128) we are getting down near the baseline probability that a particular gene possessed by A will be shared by any random individual taken from the population."The 'population' we are talking about here is the co-ethnic population. For example, English people whose grandparents on both sides were also English.
What would you think if you saw someone walking down the street greeting everyone they met with a warm smile and a cheery greeting as if they were meeting their brother or sister? I'd cross over! People at less than 1% relatedness don't stir any innate kin-altruistic feelings in us. And so to the tragedy of the NHS and other 'caring' bureaucracies. Apart from rare super-empathic individuals (aka 'saints') most staff don't have any emotional investment in their clients at all - and why should they? So the default mode is neglect - as on the street. People who create social policies are often smart, highly urbane and empathic individuals who implicitly believe the masses are just like them. Organisations, processes and laws are repeatedly designed as if no-one would ever try to game them for their own convenience. If only!
The other implication concerns your personal contribution to the gene pool. After eight generations your descendants will be no closer genetically to you than a co-ethnic stranger today in the supermarket. So stop worrying about your spurious genetic immortality! But doesn't this contradict evolution? Not really: if the human race was today undergoing strong selection (i.e. lots of people were being killed off by the 'environment' before reproducing) then as a reproducing survivor your genes would disproportionally contribute, and skew, the future gene pool: they would matter. However, in advanced countries humanity today is subject only to weak selection - the gene pool is 'stationary' - so your contribution really doesn't matter at all.
Small caveat: if you're intelligent and civilised and don't reproduce, be aware that the rather dim and less civilised are reproducing just fine: they don't have the distractions of interesting career options. This of course also steers the gene pool so it's not quite stationary.