What is not written can be very important.
And there are a natural trend to disregard what is left outside the papper.
All of the tests you mention are like:
There are 'tests' on the possible variation of 'c', 'G', 'alfa', and the test results are negative. None has achived a construction of a stable universe based in 'laws' where those constants (one at a a time ?, or combined,..) could be 'not constants'.
The exercise of varying things is pertinent because we have 3 basic physical equations and have more incognits (quantities M,L,T, and 'constants' epsilon,alfa,G,c)
The quantities are related to 'objects', say atoms, and constants are related to space (the way the space let thinghs go).
The 'tests' based on the quantities are doomed to fail, as they did, because there is something not written in the equations that matters:
We can not vary independently one or two of the quantities M,L,T, as they did in those tests. They can only vary at the same time because they are 'attached' to the 'atom scale'. It is so because it is the way we make 'rods', or rulers.
We will have a large/small 'metre' definition [L], and by 'c' constraint a larger/small time unit [T], naturally a 'large/small atom' will have a grater mass [M].
But this test was already made and claims to fit both local and cosmic data, with no break of known physical laws.
I think that those tests are mainstream physics, independent of the outcome 'fit/no fit', in particular if there is no break of laws.