Why would the fee-charging miners go out of business? Wouldn't the fee-charging and non-fee-charging miners just coexist? If one large miner decides to confirm no-fee transactions instead of fee transactions, this will not economically incentivize the rest of the miners in any way, as far as I know.
It would incent the people using the currency to not include tx fees because they know their tx will eventually get processed anyway. The more they do this, the less the fee charging miners income, and more will quit out of necessity.