You are reading too much into it. The legal question here is that is the Sherrif / County responsible when they were NOT the responding law enforcement. As I understand it, the responding officers were from LVPD.
No, they were from Metro, which is the police force for the city AND county. There is a separate North Las Vegas PD, but they have nothing to do with this.
The Department is funded by the City of Las Vegas and Clark County... Both governments must approve the annual budget including their percentage of budget. Additionally the department itself generates approximately 33% of its funds through property tax, and the charging for certain services, such as special events, work cards, and privileged license investigations. Additional funding is generated from a special sales tax to fund commissioned positions.
From wiki.
Another entity is can be formed when consolidating public services such as police and fire. They are frequently referred to as consortiums. Again, only the consortium can be sued leaving the signers of the entity free from liability.
This actually works in favor of the taxpayer by reducing insurance costs for the individual municipal entities.