You can't easily to a good job stabilizing footage if it is has a lot of motion blur (from a slower exposure) and/or if there is a lot of lens distortion. Your video has both, so there is no simple software fix for that. This is why Vegas is doing a trash job stabilizing. Post production stabilizing is really only good if you are already dealing with good footage (or ideal footage). I've worked in video editing for quite some time so I'll give you an awesome tip. Don't waste your time trying to fix that video, because even if you make it appear stable, you will not get a great video, due to the distortion, motion blur, etc.
If you really want to make it work anyway - here is what I would try:
Get some better software for this (like After Effects).
Remove/correct the lens distortion first so you are dealing with a flat image (or as close as possible).
Then create a sharper, more high contrast copy of the video.
Stabilize that copy with AE Warp Stabilizer
Then apply that stabilization correction to the first lens-distortion-corrected footage.
Then reapply lens distortion.
That is about the best way I can think of to try to fix that if you want to try, but the biggest tip I can give you - Start with good footage.