You said: "a chain of thousands of lucky mutations would be necessary before anything could be selected."
That is exactly what exists. Mutations occur daily, by the billions. Only a few are beneficial, but that is enough. The few beneficial ones become widespread over time. Therefore genetic change happens, and that is evolution. Q.E.D.
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you missed the "chain" part
what I meant was that these mutations must occur in the same direct individual family tree
A gets a mutation that will give him 0.001% of a new ability
then his son must by sheer chance get exactly a mutation that will give him 0.002%
and so on
1000 generations pass and you finally get a new ability that can be selected by the environment