Is it normal for a freestanding pergola to wobble a little bit?
I think you've got several issues:


I don't think that the single screw holding each end of the bracing in place is sufficient. 


You might consider adding a 2nd screw at each end of each diagonal brace.
I would put the 2nd screw at a different angle than the original (i.e. if the original lower screw is horizontal, add a 2nd screw at a 45° angle up). 
This may help provide some additional stiffness. You may want to try one pair to see if that helps before doing all of them. (Or wait for someone to give a more authoritative answer instead of a guess/suggestion.)

Those hurricane clips look nice, but they provide minimal stiffness along the rafter direction.


A rafter tie like this would probably work better. 
Image courtesy of Lowes.com. No endorsement implied or expressed
Yes, this one is ugly, but they make nice ones for visible work.
Adding a 2nd hurricane clip between the rafter and other joist may help improve the stiffness.

Unfortunately, I think the biggest issue is actually the most difficult to solve. I don't think that surface mounting the posts with those connectors is sufficient.


I took a look at the web-site for the brand you mentioned and they seem reasonably beefy, but they seem to be targeted more at railing posts than pergolas.
I mounted 2 short posts to my concrete steps with similar (though admittedly not as beefy) surface mounts and the posts wobbled horribly. I ended up removing them and half-lapping the post off the edge of the step and bolting it horizontally. Now it's rock solid.
I'd think you would have been better served by burying 1/3 of the post in concrete rather than trying to mount them on top. Unfortunately, this is a major rebuild at this point - hopefully someone will have a simpler solution.



Finally, it does appear from the video that there is a fair bit of wobble. How hard do you have to work to produce that wobble? Is it obviously evident from just casually leaning against a post, or do you have to put both hands on it and give it the ol' college try? 

If you have to work hard to get that wiggle going, it may be sufficient to just leave it as is. If it wobbles like that from a stiff breeze blowing, then you'll probably want to do some strengthening.