SD cards and eMMC chips[0] do have controller inside. Usually not a great one (tens to hundred-ish MB/s), but that's good enough - overwhelming majority of smartphones/tablets use eMMC for internal storage.
UFS/eUFS cards keep (sort of) same form-factor [1] but bring significant speed boost. [2]
[0] Technically, it is incorrect to say "eMMC chip", as there are typically two (or more) silicon chips inside the same plastic BGA package. One of them is controller.