The "as done by/ through" construction confused me a lot

Solution 1:

The "as" takes the exclusivity of the medium away. If we say "The performance of the children measured through testing" we are saying that we are only interested in the performance which has actually been measured by testing. If we say "...as measured through testing" we say that we are interested in the performance for its own sake but admitting that the only way we can determine the performance is by testing.

In your second example the "as" implies that, although they were interested in the type of psychedelics administered by the curanderos and their female counterparts, if they got hold of them by another route they would still try them. Without the "as" they would only be taking drugs actually supplied and administered by the curanderos/as.

Your third example is slightly different. Here the Transport Department would set new standards and determine the details of those standards but would do so because Congress required them to do so. If the "as" were to be omitted then Congress would have prescribed in detail what the standards were to be and the Transport Department would only be implementing their introduction.