Re: Expanding his diet...
DO you continue to offer it to him at mealtimes? Maybe if you put it there on his plate (or tray, however you do it) while you are all eating he'll eventually pick up on it too.
If he's eating cheese, crackers, waffles etc I wouldn't think it was a swallowing problem if he can swallow the things he likes, kwim? And if he'll eat at day care but not at home that seems to indicate he's refusing table food because he knows he can.
What sorts of things do you offer him? Maybe sweet things like fruit or berries would be an enticement to him.
“We are not put on earth for ourselves, but are placed here for each other. If you are there always for others, then in time of need, someone will be there for you.”