You must allow the weak equivalences to be unpointed homotopy equivalences. These become honest pointed homotopy equivalences between fibrant-cofibrant objects by the generalized whitehead theorem. Strom's mistake, I think, was that he didn't realize that unbased $Top$ with his model structure has the very special property that all objects are fibrant-cofibrant. This is not the case for based spaces, however (all objects are fibrant but not cofibrant). The cofibrant objects are precisely the nondegenerately based spaces.
By restricting to nondegenerately based spaces, he restricted himself to working in the category of cofibrant objects of the actual model category $*\downarrow Top$ equipped with the relative Strom model structure. It shouldn't be too surprising that this subcategory does not admit a model structure.