Out of service means the time the vehicle is occpied during the call. aka Time on Task.
We have similar services here, they only do 911 calls to the nearest hospital, then another ambulance does the transfer.
It would really depend on the harm done to the patient. Using my area as an example, any drug overdose gets transported out for treatment because our lab doesn't have the capability of determining tox levels, so, with the exception of charcoal which we can do no treatment is done here. Any substantial physical trauma is no different, they get stabilised here, then sent out. Don't get me wrong, if your local ER can do more than you can in the back of the ambulance to stabilise the patient, then take them in. However, if they've harmed themselves to the extent that they can't make it to a centre capable of treating psych patients, then your medical staff is doing a disservice to your patient by not requiring a psych assessment.
You're quite welcome.