The RV rental agencies don't know where their customers have taken on water for the fresh water tank and they are protecting themselves in case someone gets sick from bad water (I suspect it doesn't happen very often). For example, we tend to avoid well water because we don't know how often it's been tested, but city water is chlorinated and therefore safe for drinking.
I wouldn't worry about dish washing and showers, but definitely sanitize the tank. We do that when the motorhome has been sitting for a while (we keep a full tank for emergency use even when it's stored). As long as the tank is sanitized you should be able to use your onboard water for everything, the main caveat being the water may taste different in each place you fill the tank. There are many places where the water is fine to drink but we don't like the taste. I always have some bottled water available in case we get to one of those places. We like a relatively consistent flavor to our morning coffee so tend to use the bottled water for that. Also, water is my primary beverage and I'm picky about the taste so we might purchase more bottled water than others in some parts of the country. Other places have great tasting water and we use it for everything, so taste is a big factor in onboard tank usage. Once the tank is sanitized, whether you use it for drinking is purely a personal taste preference.
ArdaF