Sounds like this could be solved, if the startup could offer a higher base salary, bonus, or a combination.
Even if the current employer is big enough to negotiate lower prices or deals for its coverage and that's a barrier to entry for the startup, both companies could be set one a level field in a universal healthcare system removing this cost center from both companies. Even if the cost is (partially) rolled into something like payroll or income taxes, it would still impact the big company and the startup much more equitably.
Barriers to entry are one item this site and many of its users and are fighting to better disrupt industries and create value. If you can't get talent because you can't provide healthcare and a solution for that exists in most developed nations (universal care), why are we not engaging that?