If Elon Musk’s SpaceX racks up another successful rocket launch this week, the blastoff could go a long way toward convincing critics that the company has recovered smartly from a pair of spectacular explosions in 2015 and 2016 that cast a pall over its future.

The unmanned Falcon 9 mission from Florida’s Kennedy Space Center slated for Monday, carrying a commercial communications satellite for Inmarsat PLC, is partly intended to showcase how nimble Space Exploration Technologies Corp., as the closely held company is officially...