The basic design of Tex with commands such as
\integral pre-dates Unicode (which was first proposed in 1988, some years after the TeX82 system). The example you give (∰) U+2230 was actually in Unicode from quite early on (version 1.1, so roughly contemporary to LaTeX2e) but the majority of the standard LaTeX and AMSMath symbols were not added until Unicode 3.1 and 3.2 in 2001-2 (as part of the project that lead to the development of the stix fonts and the formalisation of the entity names in MathML and later HTML5).
Even today, and even if you are using xetex or luatex (which can use the full Unicode character set natively) many people find it more convenient to use an ascii markup form than access the characters directly. It is easier to type on most keyboards and is self documenting. If you look at the source of a document and see ⨌ then short of cutting and pasting a fragment and hoping, it is hard to know how to generate that symbol, whereas if you see
\iiiint then it is fairly clear that you just need to type
\ i i i i n t.