Is the in home care agency responsible if my mother's car was totaled in an accident if the driver was the caregiver?

Are you in a no fault state? No fault means mom's insurance covers injuries to mom. As for the collision coverage, it depends who was at fault in the accident. If it was the other car's fault, the other car insurance pays. In some cases the fault is not readily determined, so each car is covered by its own insurance. If the caregiver was injured, that becomes a workman's comp claim since she was on duty.

The agency is responsible for the caregiver being a good caregiver and not abusing or denying the patients care. I was bonded and my background checked. The agency is responsible to show the family (or whomever is now responsible for the patient) that the time spent in care is worth what they're paying. We kept logs, everyday of what we did, what we spent, where we went, my mileage, if my car was used, the client's mood and whether or not they'd taken their meds--think about a nurse charting a patient...very similar. I had to account for my time.
IF there had been accusations of abuse or neglect, my company would have been "responsible" in some ways--although the burden of care lay on me.
There was a contract signed by the family and by me, stating I would do this and such and they could expect this and such in return for my time.
In my case, I was NOT required to help my client bathe. Some other clients, maybe, but mine didn't want that.
Agencies protect both sides of the coin, so to speak. Also, if I called in sick (never did!) the agency was responsible to send in a replacement.

What is the agency responsible for then other than placing the caregiver in the home? Anything?

I worked in Elder Care. I drove my client a LOT. I was required to show proof of insurance, as was she. Her CAR was insured, she no longer drove.
My agency would not have covered any kind of accident. My insurance would have, after my $500 deductible, if the accident was my fault. If it wasn't, then, the other person's insurance would have been involved and that would not be my problem.

I believe the driver's insurance covers the damage, not the agency. When I was a home care nurse we were required to show proof of automobile insurance for this purpose specifically.

Probably not but it would be spelled out in the agency's agreement

And BTW, I hope no one was badly hurt!

There may be some personal responsibility if the caregiver was charged as a result of the collision, it gets murkier if you want to connect it back to the agency. This is why most agencies here won't allow their caregivers to get behind the wheel without a signed waiver and proof of adequate insurance.