It is difficult when your 2 parents, both aging, do not agree on how and where to age. The best solution is to look a solution that has something that
pleases everyone, including you.
First, please check with your local nursing homes. I've not heard of vets getting free care. There is a veterans home near me in TX, and the fees are reduced, but there is a long waiting period and they are not free, just less expensive. Other nursing home do not have special pricing for vets that I am aware of.
There is a program called Aid and Attendance' that provides some financial assistance, http://www.veteranaid.org/program.php.
Around the clock in-home care will cost a minimum of $300 per day. If your father can afford that, then he can afford care in a nursing home of his choosing, near his home if that is the best choice. Having him in a care facility gives your mother her desire not to have her privacy violated and also probably offers as good care as he would receive in-home. Your family and mother and friends can visit and provide him quality time, which trained professionals tend to his physical needs.
Another option is for your mother to temporarily move to a senior apartment, while your father has in-home care. She can move back home after his death. This solution also meets her desire not to have strangers in her home.