Why would God make these allowances for His priests? What does this reveal about His character towards those He calls?
The exceptions God makes in Leviticus 21:2 are described as a 'condescension to human infirmity.' This means God, in His wisdom and grace, recognized the natural, powerful emotions that arise when we lose loved ones.
Even though priests were set apart for holy service, they were still people. The grief of losing a parent, child, or sibling is a deep, primal human experience. To expect them to suppress this entirely, except for these very closest bonds, would have been excessively harsh and perhaps even unrealistic.
These allowances show us a God who is not distant and unfeeling. He is a God who understands our makeup, our capacity for love and sorrow, and who builds His expectations around our real human experience, guiding us even in our grief.