This is a sad discussion. MO does not have a leash law but pet owners are legally required to reasonably restrain their pets so they do not endanger others. Willy could have been hurt or even killed. Better the dog suffer than he. The question raised here, however, appears to be what was Willy's responsibility after the accident / striking the dog? I suppose you can argue about his moral responsibility but I think the answer is quite obvious and consistent with the law. A dog is personal property under MO law. Thus, that question is resolved by the MO Hit-and-Run statute:
Leaving the scene of a motor vehicle accident.
577.060. 1. A person commits the crime of leaving the scene of a motor vehicle accident when being the operator or driver of a vehicle on the highway or on any publicly or privately owned parking lot or parking facility generally open for use by the public and knowing that an injury has been caused to a person or damage has been caused to property, due to his culpability or to accident, he leaves the place of the injury, damage or accident without stopping and giving his name, residence, including city and street number, motor vehicle number and driver's license number, if any, to the injured party or to a police officer, or if no police officer is in the vicinity, then to the nearest police station or judicial officer.