I don't think it's about being actually able to send a message. It's having the transfer of dollars recognized. Say I send $50 million from my account at Sberbank to UBS and then try to transmit from UBS to my boat builder in the Netherlands. That won't work because UBS won't get $50 million added to their balance sheet because other banks don't recognize the Sberbank -> UBS transfer since it's not on SWIFT. To them, UBS will be trying to spend money it doesn't have and the transfers will get rejected.
In short, No SWIFT = No Dollar transfers outside the country.