Masking tools in Warp Stablizer (to decide which parts of a clip get analyzed!)
The Warp Stabilizer tool should have built-in masking tools for users to be able to decide which part(s) of a clip get analyzed for stabilization! (i.e. ability to ignore from analysis certain moving objects that can throw stabilization out of whack.)
TIP You can already decide which part(s) of a clip get analyzed for stabilization!
But it takes a few steps:
1. Mask out parts of the clip you want Warp Stablizer to ignore using Opacity effect mask(s).
2. Nest masked clip.
3. Apply Warp Stabilizer to nest.
4. Once the clip is stabilized, open nest.
5. Turn off Opacity effect so entire clip can be seen again.
6. Exit nest.
Voilá! Near After Effects-level stabilization inside of Premiere Pro!
If Warp Stabilizer had built-in masking tools, those 6 steps above could be reduced to only 2 steps! Plus, this trick would be a lot more discoverable. As a bonus, it could potentially allow WS to analyze masked clips quicker, since it would have fewer moving pixels to analyze! This would be a win-win for users!

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Pierre Louis Beranek commented
Here's a VIDEO DEMO of this concept in action: https://youtu.be/vDpZK3ruET0. Please share it with anyone you know who could benefit from this currently working stabilization trick!
Correction: if Warp Stabilizer had built-in masking tools, it would only take ONE vs 6 extra steps to get AE-style selective motion analysis! (not 2 as mentioned above) :)