Picking a different column in each row of a Dataset[]

With Dataset's syntax sugar you may implicitly use Query.

ds as in OP, then

ds[
 GroupBy["x"],
 All,
 KeyDrop["x"] /*
  Normal /*
  Curry[SelectFirst, {3, 1, 2}][FreeQ[_Missing], "x" -> Missing[]]
 ] //Normal

<|1 -> {"b" -> 10, "x" -> Missing[], "a" -> 11}, 2 -> {"a" -> 12}|>

Hope this helps.

foo = Select[MatchQ[Except[_Missing]]] /* 
      (# /.  Association[] -> Association["x" ->  Missing[]] &) /* 
      KeyValueMap[Rule] /* 
      First;

ds2 = ds[Select[#x == 1 &], {"a", "b", "c"} /* <|"fc" -> foo|>];

Join[ds[Select[#x == 1 &]], ds2, 2]