How to replace loose 4x4 wood fence post with a steel post
Have you tried tightening the nut on the bolt that holds the post to the saddle anchor?
I had a very similar setup though my saddle had two lag screws.  My post was quite wobbly so I pulled the lag and drilled through the saddle to get a setup like yours.  Once I tightened the saddle to the post it was quite a bit more stable.  My setup has two bolts and nuts.  If tightening the one in your saddle doesn't work then you might want to consider drilling the metal and adding an extra bolt and nut.
Since the legs of the saddle are facing the direction of swing for the gate the force should be put against the saddle.  If you wobble the post does the saddle wobble or is the post wobbling within the saddle?
If the saddle wobbles then I'd chop it off and drill a new hole, epoxy a new saddle in place and put a new post in.  You can get heavier duty saddles or you could even add lengths of plate steel (choose a sufficiently thick piece and bolt it to the saddle to extend the vertical resistance).
Since the saddle wobbles I'd remove it.  I've attached an image of my setup.  This post does not wobble at all in the direction of the saddle legs.  You can drill into the concrete and epoxy threaded bar to secure the saddle to the concrete.  I didn't originally install this saddle but it seems to me it was installed after the concrete pathway so I'd bet my saddle is only secured by wedge anchors and they appear to be doing a fine job 25 years on.