Not everywhere. For example, Wisconsin CS is easy to declare (earn a C in a CS course) after enrolling as undeclared, so it is not likely that direct admission to CS will be harder than admission to the school, unless direct admission is not offered (but then it is not really a problem, since it is easy to declare CS after enrolling).
https://www.cs.wisc.edu/academics/undergraduate-programs/how-prepare-cs-major
If one starts looking at (numerous) less selective schools, many of them do not have CS as more competitive than the school overall. For example, of the 22 CSUs that offer CS, only 9 of them have CS listed as “impacted” (i.e. more competitive than the school overall), leaving 13 where CS is no more competitive than the school overall.
https://www.calstate.edu/sas/documents/impactedprogramsmatrix.pdf
Of course, even at some schools where CS is more competitive than the school overall, it may not be so competitive that a student for whom Harvard and the like are realistic (as opposed to unrealistic) reaches would find them too difficult to get into for CS.