You are correct. It is just like a strike. You need leverage. The leverage is no hike of the limit. Give that up, then you give up your leverage.
The concern is not paying out some of the other goodies such as social security, covering costs of securing the country (e.g. the important parts of the military etc.) while paying the interest on the debt. That could be a serious political problem.
Time for a RICO case to be brought against Lew, DOI and the NPS. Extorting the citizenry for funding.