It sounds as if your mum is 86, and reasonably well apart from the deafness and dizzy spells. I would just open the conversation with her. Explain that you get worried, that you know that she can look after herself but that her hearing means that it sometimes hard for you to get in touch when you are concerned.
You could go on to talk about how you can decide on the limits to preserve her privacy for eg using the bathroom and getting dressed/undressed. Perhaps you can agree to check on her set times/frequencies rather than watching her the whole time if she feels that that would be intrusive. I think you’d have to be disciplined about not making comments about things you have seen! And tell her that if she is really uncomfortable with them, then you’ll turn them off.
She can only say yes or no. Sometimes my mum says no to something the first time and then mulls it over for a couple of weeks and decides it’s a good idea. I hadn’t thought about this as an option for her, but I’m starting to.
Try to think of how you can present this as a benefit to her? If she needed help she could call you via the sound camera….the ones I see F&F use for pets/babies have two way sound. She may not be able to hear your response, but she would have the confidence of knowing that you are responding.
DD and I don’t have location tracker set on our phones, and lots of our friends think this is a bit odd! We are used to being independent. I was in a controlling relationship for years and the very thought of this makes shivers run down my spine. But I never say never - if I thought that eventually this kind of tech would keep me safely in my own home for longer, and give my loved ones peace of mind, I would be prepared to give it a go.
You might not want to go to the cost, but technology might help you with the door as well. If it’s the front door, a ring doorbell linked to your phone will tell you if anybody tries to go in who shouldn’t. But there are keyless electronic alarm/locking systems now which you can operate remotely.
Finally, you’ve probably been down this route - but a couple of people I know who suffered with poor hearing for years have finally found Bluetooth hearing aids that work for them, especially with the phone, which has been a huge source of frustration.