Why are there so many restrictions on /ŋ/ in English?
Most of the unusual behavior of [ŋ] can be explained in its historical origin from simplification of the consonant cluster [ŋg] (in words where [ŋ] occurs in modern English outside of the clusters [ŋg] and [ŋk]).
Compare the behavior of the consonant cluster [mp]: like [ŋ], [mp] cannot occur at the start of a syllable and rarely occurs after a diphthong or "tense" vowel.
The clusters [nd] and [nt] can be found after diphthongs more frequently than [mp] or [ŋ], but that is part of a general pattern where syllables ending in coronal consonants such as [d], [t], [s], [z] are allowed to have more complicated structure than syllables ending in non-coronal consonants.