I notice some insurers offer the option to buy two or more consecutive short-term policies with no medical underwriting required after the first application. So does that mean I’d be covered if I buy a sequence of short-term policies and then get sick?

Probably not.  Each of the consecutive policies that you would buy would likely include a pre-existing condition exclusion.  So, for example, if you are healthy enough to buy two consecutive short-term policies today, and then you get cancer while covered under the first policy, your cancer would be considered a pre-existing condition when the second policy starts and so would be excluded from coverage under the second policy.

By contrast, a qualified health plan sold through the Marketplace will never exclude pre-existing conditions.