...instead of just turning himself into a bird? I mean, he turned himself into a fly before.
This is something very usual to happen in old stories, specially in the mythological ones. They have a dream-like logic and a dream-like inconsistency. The latter is clear, for example, when talking about sizes, Fenrir may be a normal sized wolf when tied down by the æsir, but he is sometimes described as having a jaw that can touch the sky when open and so on.
So, the fact that Loki needs to take Freyja's coat has no special reason rather than to tell a story with underlying meanings and making it reasonable. Why would Þórr just kick Litr into the pyre where they were burning Baldr? Why would Sigurðr say to Fáfnir that he wouldn't tell him his name nor his father's name and then right after tell these things to Fáfnir?
It has no right answer now (there could be an old lost myth where it is explained why Loki needs to take it or why can't he just turn into a bird right away, or maybe Loki never had shape shifting abilities in some traditions and had to borrow magical things to do so), and, unfortunately, it may never have had a clear reason.