If the light was "stretched," ie, speeding up, in order to remain constant to the expanding space, it wouldn't be redshifted.

It is the assumption that space is expanding, relative to the speed of the light and so that light takes longer to cross, is the reason for the redshift.

Classic Doppler shift doesn't mean the space expands, only that the distance increases. The train moving down the tracks doesn't stretch the tracks, it only increases the distance. In this case, the speed of light is still being assumed as the tracks.

Which makes the speed the "ruler," not the expansion. That is what is being measured by the ruler.

A popular analogy that is used is of an inchworm, as the light, crawling on an expanding balloon, as the space. The expansion of the balloon doesn’t speed up the inchworm. It is the difference between these two, that is the basis of the redshift.