There's a great study that shows how the "homeboys" driving their buddies to the hospital after gunshot / stabbing trauma have better outcomes than the paramedics because (1) the police mostly delay paramedics' arrival, (2) homeboys are already on the scene, (3) homeboys are faster at getting to the hospital. This is why police are not allowed inside hospitals to interview patients - their loitering outside ERs is barely tolerated by staff in major metros. They harm health outcomes (surprise, surprise).
Is this lack of training? Where were the police earlier, before the gangsters shot each other? There is already evidence they delay care and harm outcomes, and they do not change what they do, so why would they carry life saving devices? They don't, because like with drug addiction, public health isn't commensurable with modern law enforcement.
On the flip side, nobody is really satisfied with the idea that it takes a trauma surgeon to save you. You're right that it takes skills, but not skills that a paramedic, a nurse or many other people have.