I wonder if a tablet or laptop might be more appropriate for your wife? When I think “web” I want a keyboard or touch screen and a TV typically has neither. But in any case, I think you’re making a mistake trying to have one device that can fit every use case. Your TV will have multiple inputs (and it will also probably be a smart TV).
Plug some sort of Mini PC into the TV for your wife, and let your kids use the TV’s built in smart features to watch TV or buy a set top box such as an Apple TV/Nvidia Shield/etc.
PS: I would 100% use a projector, not a TV. Just project onto the wall (assuming you don’t have wallpaper/etc).