Most families wait for the fall, because a fall is the only thing that feels like proof. The walk gives warning well before that. You do not need Dad to admit anything for you to see it. You only need to watch him cross the room.
Local option: New Plan Care gives families in the St. Louis area in-home help so a parent can stay home: companionship, help with bathing and dressing, meals, and overnight care. Call (314) 405-0887.
What to watch for
Pick two routes he uses every day, such as the chair to the kitchen and the bedroom to the bathroom. Watch from a distance so he is not performing for you. Look for a hand that touches the wall or a chair back at each doorway, feet that drag or slide near rug edges, shorter steps with the feet kept close together, a long pause before turning or sitting, and a grip on a rail that he did not use before. Write down what you saw and roughly when, in plain words. One bad morning is noise. The same change three times in a row is a pattern.
What this does not tell you
Watching the walk does not tell you why it changed. That question belongs to his doctor, and you cannot answer it from the driveway. Your job is to bring the doctor a clear record so the visit does not turn into a guess. Before the next appointment, write three lines: what changed, where it shows up in the house, and what you want checked. Ask the doctor directly whether his walking or balance has changed since the last visit and what they would want to look at.
Test the house against the walk
Walk his route yourself, at the hour he uses it most, with the lights he would normally use and nothing extra switched on. Check the lights on the route at night, rug edges and cords, the step into the bathroom, the seat height of the chair he uses to stand, and whether a grab bar or rail sits where his hand goes. Where his hand already lands is where a fall will happen if that spot is not supported. Dad may be right that the house works for him as it stands. The way to test that is to see where his hand goes, not to argue about it.
Compare the walk across the day
Compare how he moves in the morning with how he moves in the evening, and after a meal or a bad night with a good one. If the walk is only worse at one time of day, the fix is often a route change, a light or a seat, not a new plan for his whole life. If it is worse at every time of day, take that to the doctor first.
The next move tonight
Tonight, walk the bed-to-bathroom route with the lights he would use at night, mark every place his hand touches a surface, and tape down any rug edge you find. Then write the three lines for the doctor and leave them where you will see them before the next appointment. If you need a starting list of local people who can look at the route with you, see Local Help in St. Louis.