Why is 台北 spelled Taipei, not Taibei?
From wiki: the spelling "Taipei" derives from the Wade–Giles romanization T'ai-pei. "Taibei" is pinyin, which iOS supports as an input method.

Wade-Giles used to be the standard method of romanization so it shows up in a lot of older Chinese names like "Chiang Ching-kuo".

Pinyin has its own disadvantages, especially when spoken by English speakers. Consonants like "zh", "q", or syllables like "ri", "shi" sound very different from what an English speaker might expect.