The President has no necessary role in the Constitutional amendment process. Informing people that he had no objection to it was effectively permission to support it for his party followers.
Corwin mattered in Maryland and especially in Kentucky, both of which states had lots of Democrats but nearly zero Republicans.
Unionist Democrats liked Corwin, Republicans did not, but went along with Democrats in hopes of preserving the Union.
In the end Corwin may have helped keep Kentucky & Maryland Union, but otherwise had no affect.