Although you may not have had the care from your Dad's GP that you wanted or expected, Tilly, from your post, it sounds as though the GP has done what was needed for your Dad.
He listened to the care home staff and immediately issued a prescription for antibiotics for his chest infection. I have no idea of the mental health services in your area, but going to A&E is sometimes the quickest way of being seen by a
Mental Health Specialist.
As others have said, if you feel aggrieved, by all means make a complaint.
I do not want to sound harsh when you are clearly concerned about your Dad and probably also rather worn down by caring for him, but try to see the 'big picture'. He has had the treatment he requires for his chest.
You might , however, want to ask questions of the Care Home staff as to why they had not spotted his chest infection.