Nina, in home care givers do not have the expertise and availability of medical support personnel to decide what is important or life threatening. They will tend to send a loved one to be checked out for minor symptoms just in case. Also your FIL was much earlier in the disease.
In a care facility there are usually doctors and nurses to assess the situation. If they can handle whatever is going on they do not transport the residents to the hospital. But if the patient become critical or they believe there is a good reason then they do send them to the hospital.
When you make the decision to go with Palliative care it changes dramatically. At that point you are concerned with comfort. Most dementia patients do not die of dementia. The dementia does deteriorate the body, decrease the immune system, and limit the ability to recover. It is normally some other complication that takes their life (pneumonia, heart attack, stroke... and many other causes). At the point of Palliative care you keep them pain free and let the progression of the dementia and other complications take their course. You may try antibiotics to cure an infection or use oxygen in place for breathing difficulties... but these are also comfort measures. The facility can do as much as the hospital can.
Dad had vascular dementia. He also had chronic heart disease and some other medical problems. For a while he would go to the heart doctor, have stents, etc. Then the family, along with his cardiologist, decided it was time to stop the aggressive treatment. He did not understand that he needed to lay still after the last stint and bled out. It was scary and what lead us to decide... no more! Other problems were address over the next two years. He had antibiotics for a UTI. He had PT for his arthritic knees. He was in the ER for falls. Then the decision was made (mainly because of the trauma of each ER visit) that we would go with Palliative care. From that date he never did go back to the hospital. I am not sure if he died of heart failure or kidney failure but it was peaceful, in his familiar surroundings of the facility, with his family around him... including Mom.
SO have peace in what was done. Perhaps it was sooner than you expected but again... they chose the time. I never imagined that Dad would live over 2 years after the cardiologist released him but he did