On a step of proving (geodesics defined for all time)=>(for any $p,q$ there exists a minimizing geodesic connecting both), besides the traditional approach of taking smooth curves $c_n$ such that $L(c_n) \to d(p,q)$, Sakai mentions that one can use Arzela-Ascoli (changing the assumption of geodesics converging for all time to bounded closed balls being compact) to show that there exists a subsequence of those curves converging to a continuous path $c$. By continuity, we have that this path is length-minimizing. Therefore, it is a geodesic. However, I feel that the result that a continuous length-minimizing path must be a geodesic via the series of exercises and lemmata which are mentioned is very technical and kind of goes against the purpose of the argument of being cleaner.
My question is: is there a way to make a shortcut here via some regularity theorem? For example, maybe arrive somehow at the fact that the limit is a weak solution to the geodesic equation and therefore must be smooth? (I don't know any references which approch the subject in this way*, so this would be nice too, if possible).
*Except Klingenberg's, but his intentions are others.