This is an interesting idea. I think Chrome OS sheriffs have previous propose the idea fo an "investigating" button which would be in a similar vein. 

I think this wouldn't be a bad thing to have though I'd be concerned that we might be adding too many actions to the UI, which will make things a bit more complex. I think before we make too many small workflow changes, we should try to think some about what we want the ideal Sheriffing workflow to look like, so that we don't make a bunch of different little UI changes that pull Sheriff-o-Matic in different directions. At some point, I want to write up a design doc on different ways we could improve the Sheriff-o-Matic workflow and work with sheriffs to figure out an ideal workflow shape for the tool that works well for different teams. 

For example, with an "I've got this button", I imagine SoM might have much more of a workflow where sheriffs would essentially assign issues to themselves, then once sheriffs have those issues assigned, they would probably spend most of their time looking at their assigned issues or other unassigned issues.