Your assumption is not foolish at all, you can customize almost all standard JHeadstart behavior to satisfy specifc requirements.
For this situation, you want to customize the templates that generate the LOV task flow.
In this task flow, you should have the main router first navigate to a new method activity that invokes the jhsSearchBean.clearSearchCriteria method, and then navigates to the LOV page itself.
It is easiest to first get this to work by modifying the generated LOV task flow, and then move your changes to the relevant templates to preserve the changes upon regeneration.
See chapter 12 of the dev guide for more info and tips on customizations.
Have tried to implement your suggestion and it seems like what you describe is nearly the default behavior. The big difference is that the method called after the main router is clearSearchCriteriaAndRequery. Changing this to clearSearchCriteria doesn't seem to have any impact on the QBE, and both methods seem to impact only the query panel and not the QBE.
I've also asked the question in the JDev forums, because I think what I need is the ability to access the popup's equivalent to a Filter Model. Does something like this exist?