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

Question

All these sentences with "as done..." format have been confused me because why not just omitted "as" in these sentences?

Mr. Miller has said he does not support a mandated federal test but
does favor public reports on how college students are learning as
measured through testing.
The New York Times

It seems to me that "...how college students are learning measured through testing" is enough to express the meaning of it, and it meets my perception of English grammar. Why is there an as?

With his wife he went to Mexico to sample psychedelics at their
practical source, as administered by the curanderos and curanderas of
the Sierra Mazateca.
The New York Times - Magazine

"With his wife he went to Mexico to sample psychedelics administered by the curanderos and curanderas of the Sierra Mazateca at their practical source."Can this sentence be written like this? It seems to me it's a structure used to put clause used to moderate objective in end of the sentence, with an "as" leading the clause.

And the Transportation Department would set new vehicle mileage
standards as required by Congress.
The New York Times

We could've just make the sentence "...new vehicle mileage standards required by Congress". Could haven't we?

BoldBen · Answer

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.

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

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