Yes the final Progressive objective is to get rid of private health insurance.
The fine is so small that the government won't make much from it, and the bill specifically forbids the IRS from taking punitive action for NOT paying it, unlike the income tax.
So as the insurance companies get raked they either raise their rates or drop the whole business, making things worse.
This will lead to public support for further expanding medicaid as I said, AND making the mandate tax bigger and more punitive as those not insured will either be demonized or made into victims of the greedy companies depending on income.