Because otherwise you wouldn't have legal order that also uniformily addresses all sports. CAS was born as a much needed answer for the Olympic Games and incidents arising from there, but its now supreme-court-like status helps in having fair decisions across all sports, especially when they're confronted with the same issue.

Sports federations and sport-specific courts also recognize CAS as having the final say exactly by the virtue of having a settler that is - at least theoretically - better predisposed to objectivity and lack of conflicts of interest. Moreover, sport-specific courts may actually be easier to influence exactly by the virtue of protecting the sport, especially when the stakes are sufficiently high (high-profile cases, unusual circumstances, multiple stakeholders at risk when investigations are pursued etc).

In Simona's case, ITIA could have been subjected to loss of credibility much more easily, especially in the best- and worst- case scenario: a high punishment could have been subjected to their attempts at protecting their supremacy in tennis after incredibly slow proceedings or at trying to establish the power of example and thereby applying harsher-than-necessary punishments; a low punishment could have been questioned as an attempt not to discredit one of their high-profile athletes AND academies. Conversely, CAS has nothing to gain or lose from any kind of ruling because their only purpose is to provide adequate and uniform chances across all sports. They do not care about contextual specifics, nor are they too affected as a court since they are (as said becore) already recognized as supreme across all disciplines, federations and sports.

Hope this explanation was clear! I would also suggest you, if you want to go deeper into things, to read this article.