Have you tried first doing the constraints containing x, y, and z separately? For example:

In[9]:= Reduce[
 Join[system[[{1, 2, 10}]], {0 <= \[Omega]1 <= 1}]   , Reals]

Out[9]= (0 <= \[Omega]1 <= 1/Sqrt[2] && -(1/2) < x < 1/
    2 && -Sqrt[((-1 - 2 x)/(-3 + 2 x + 4 \[Omega]1^2))] < \[Theta] < 
    Sqrt[(-1 - 2 x)/(-3 + 2 x + 4 \[Omega]1^2)]) || (1/Sqrt[
    2] < \[Omega]1 < 
    1 && ((-(1/2) < x < 
        1/2 (3 - 4 \[Omega]1^2) && -Sqrt[((-1 - 2 x)/(-3 + 2 x + 
          4 \[Omega]1^2))] < \[Theta] < 
        Sqrt[(-1 - 2 x)/(-3 + 2 x + 4 \[Omega]1^2)]) || 
     1/2 (3 - 4 \[Omega]1^2) <= x < 1/2)) || (\[Omega]1 == 
    1 && -(1/2) < x < 1/2)