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I can't speak to Stravinsky specifically, so perhaps this is not an answer to your question, but you also ask:

> How could a skilled composer like him be unskilled at conducting?

Composing, playing, and conducting are different (related) skills.  A person can be an excellent performer, including being skilled in all of the finer points of tempo, phrasing, shaping, and more, and yet not be a good *leader* who can unify an entire orchestra.  I've seen this (at the amateur level, admittedly) with choirs; a person can be an excellent singer, spot-on on pitch and tempo and with good blend, but not have the "topsight" and the ear for conducting the whole.  Conducting involves different skills, separately taught in music programs.

Being an outstanding performer or an outstanding composer gives one an advantage, over those who are not, in conducting.  But it's not automatic.  A person who invests in cultivating those additional skills will excel as a conductor; one who doesn't probably will not.

This answer is based on personal experience and observation, mostly choral and not orchestral.