The rock-splitting contest is exploring Elrond's relationship with the elves. It isn't about smashing rocks. It's about Elrond submitting to a Dwarvish custom

The rock-splitting contest is exploring Elrond's relationship with the elves. It isn't about smashing rocks. It's about Elrond submitting to a Dwarvish custom and making a not-terrible showing (for an elf). It demonstrates the respect he has for them and shows the half-measure of respect he earns from them. That is the key scene of the episode. The reconciliation would not have been possible without it.