I think you'll find if you do a little research that Republicans use the idea of trickling down to justify their tax cuts. They even make the CBO change their accurate models to "dynamic scoring" models to help sell those tax cuts.
No, "trickle down" is just what the opposition called it.
"Political opponents of the Reagan administration soon seized on this language in an effort to brand the administration as caring only about the wealthy."
The argument was never that "tax cuts for the wealthy make it down to the middle class", it was supply side - lowering taxes gives people more money (in general) which stimulates economy activity.
