I've found the opposite... the higher you go in math, the less useful arithmetic skills are, and the more critical it is to be able to think logically. I regularly tutor high school kids in Calc II and Stats who don't know their multiplication facts from memory, but they all have calculators so it's never an issue...
The higher you go in math, the MORE useful both arithmetic AND logical thinking are. One does not cancel out the other. You need both. Not knowing your basics prevents you from logically contemplating/exploring the mathematical topic at hand.
(Calc and Stats are considered an extension of basic math, by the way...higher math includes linear algebra, complexity of vector spaces, mathematical biology, etc...that was considered the case at both my Ivy grad school and my very un-Ivy undergrad institution).