Heavy crude does need additional processing equipment and procedures at greater cost. The main issue now is that refineries built only for light sweet crude (like many US refineries) cannot process heavy crude at all. Even if these refineries wanted to process heavy crude, they couldn't do it, and it would take years to add in the new  equipment. I think it's capacity, not extra costs, that is the big issue right now.
Were most of the refineris closed in USA sweet-light refineries?

Perhaps the refinery capacity cut in USA has not made any difference, it might even have been beneficial if it has made it easier to invest in equipment for handling heavy and sour crude.