Col Brandon has always spoken quite freely with Elinor on intimate matters.

There was that occasion at Barton cottage when he speaks about Marianne and youth etc.
Then there is the time when he tells her about Willoughby.

I think it most natural that he would approach her for something else which is so very intimate.

If we take all the common acquaintances one by one we'll see that no one qualifies.
As others have said going to Edward personally was not a good idea.
It could embarrass Edward.
It would be difficult for him to explain why he was doing it.
It could be that Col Brandon wants to do a good deed without being in the limelight.
And they didn't know each other so well to be doing it personally.