Ah, so M1 sounds roughly equivilant to what we call a 5150 here in CA....here in LA there's a large pool of IFT providers (what, something like 80+ ambulance companies that do exclusive IFT work here?) If it's a cooperative in patient IFT there's no reason to pull a cop out of service. The first private company I worked for didn't have any special psych patient restraint policy on top of the normal existing restraints policy, while the other private I worked for mandated minimum two point restraints on all 5150 patients regardless.
For 911 calls, it really all depends. We'll call PD if the patient is violent or combative (though hopefully if dispatch gets information the patient may be that way they'll auto dispatch PD alongside us, and we'll usually stage out until they have control over the scene). From there it really all depends on things like how "sick" the patient is, their vital signs, their history (new onset? or a flare up of a known existing issue?), what exactly the patient did/didn't do or said and whatnot, basically boiling down to what do the cops and medics on scene think? If everyone agrees that the patient doesn't really need an ambulance ride than we're more than happy to let PD handle it, and if the cops or medics feel more comfortable with ambulance transport that's what we'll do, usually with either an officer and/or FF riding in back as well.
There's been a couple calls I've been on where the patient needed chemical restraints, and that's no question an ALS ambulance transport here (PD riding in back as well with additional officers driving right behind the ambulance). One time after staging out for PD we arrived on scene to find the violent, combative psych patient restrained in the back of the police car, still so agitated no one wanted to wrestle him out of the back and onto the gurney. PD still wanted him checked out at the hospital, and after a bit of discussion, we all decided to basically convoy to the hospital, the fire engine, our ambulance, the police car with the patient/suspect still handcuffed in the backseat followed by the other 3 or 4 police units that had responded. Basic idea being if something happened to the patient enroute we'd all stop, do what we needed to do to transfer the patient into the ambulance and go from there, but fortunately the convoy was fairly uneventful and nothing untoward happened and the patient had calmed down enough to be walked into the ER by the police so we were able to clear right away. Definitely one of those one-off, wont see that happen again (at least for a while lol) calls