I don't understand your last point, Francis
If a person did believe, as you suggest, a woman should terminate because of a diagnosis of DS, earlier testing would be an advantage (not disadvantage) as earlier terminations are less complicated.
If you believe a diagnosis of Down's has no impact on a woman's decision to terminate, testing early is potentially an advantage as there would be more time to find out about Down's and make preparations for caring for a Down's baby.
The disadvantage to the test(as mooted as an implication by the documentary) is that, if you believe a woman should continue the pregnancy in the case of a positive test for Down's, she might terminate (due to testing positive for Down's), when she would have continued with the pregnancy and refused the amnio because of the risks associated.
Another disadvantage, I personally wondered about, was that women could be deciding to terminate based on less information, regarding the actual health condition of a foetus testing positive for Down's. Because an amnio test is given later more can be determined concerning the feotus' development. Some women might want to continue with a pregnancy in the case of Down's if there were no very severe health conditions detected at that later stage. Although, of course, late abortion is more complicated and traumatic.
Just because disadvantages are spotted does not mean the test should not be offered IMO. More that there should be no attempt to persuade a woman to take the test when she has declined it.