I have a couple of suggestions.
If you are waiting for him to realise that he has dementia or trying to get him to understand or admit that he has then you are fighting a losing battle. Even if you tried and tried with him and he finally said that yes, he knew he wasn't ok any more and that you should take over what he had been doing, (and it is far more likely that it will end with a huge argument and storming out) chances are that the next day or even sooner he would have forgotten again.
Don't worry what your sister says, you can't just "take charge" it needs a bit of subterfuge.
Do you have power of attorney, that is really important. If not, it may be easiest if you let him think you are worried in case you get ill and he needs to have POA for you, and while you are at it you can do the forms for both of you, that's the sort of subterfuge I am thinking of here.
Try to intercept post so that you know what is going on and see if you can make it a gradual process. If there are any specific things that you need someone to help with it I am sure you will get some good answers here.