Most people do not have the luxury of choosing between nursing care in a facility and private in-home nursing care. While nursing home care is expensive, averging around $6,000 a
month, that is cheaper than in-home nursing care. Private duty nurses charge around $100/hour. In-home health care agencies generally have a nurse on call or a nurse who may come to the home for a few hours a week but the majority of around the clock care is supplied by a health aide or a Certified Nurse Assistant. This option is very expensive. A less expensive option is to have in-home care for several hours a day rather than around-the-clock.
Part of the answer to your first question centers around how much care by a nurse is needed. If your mother's health situation is acute, then being in a nursing home is the best option to have a nurse nearby. If she needs a nurse only to supervise the day-to-day care from an aide, then in-home care becomes an option, if you can afford it.
The majority of people in nursing homes come in after a hospitalization. They may stay a while in the rehab unit and then transfer back home or to a long term stay unit in the nursing facility. My point is that the patients in a nursing home are generally sick enough or weak enough to need around the clock care when they come in, and can expect their health to improve as they recuperate.
You stated that you do not like the 'level of engagement' in the nursing facility. Do you refer to the nursing staff being engaged with patient care, or to the ability of the activities director to engage the patients' interest in something outside of their physical discomfort? While activities in a nursing facility may not be great, life in a nursing home in less isolating than life in a private home. At least in the facility there are things going on, people doing things around them, and there are other patients to watch or talk to. In a private home, the patient is alone. This frequently leads to depression and paranoia.
The last part of your question deals with 'letting nature take its course.' Sometimes this is called AND: Allow Natural Death. This is appropriate when the patient has no quality of life and no expectations that meaningful life can be restored. Death happens at home and in nursing facilities.
I recommend that you have a discussion with your mother about Hospice care and about end-of-life decisions. Talk with your clergy, with trusted friends, with other family members, and with your mother's physician. Ask whether her quality of life can be recovered, ask her what she needs to have in order to make her life meaningful, and then make a decision based on facts and her wishes about letting nature take its course.