Not quite right. The GVWR is a maximum limit.
The weight from the heavier tires will reduce the cargo capacity.
Example: My '06 with me and a full tank tip the scales just shy of 4500 lbs. The cargo capacity of my truck is 1205 lbs (or 1/2 ton). So, the GVWR of my truck is around 5700 lbs. If I install heavier tires, say 200 lbs (heavier than stock), the empty truck with me and a full tank will be about 4700 lbs. Cargo capacity will be reduced by the 200 lbs, but the GVWR remains at 5700 lb.
I know the weight of my truck due to many trips over the landfill scales to dump construction debris. I also know 1300 lbs will compress the rear springs by 2 inches. Yes, I overloaded a bit.
Similar math when considering the tow rating. My truck has a rating of 6500 lbs for trailer laden with cargo. It the tongue weight is 10-15% of the trailer, then the tongue weight is 650 lbs. This load is considered "on the truck" and is counted against the Cargo Capacity. Using 1205 lb cargo capacity of my truck, the tongue weight leaves me with 555 lbs for people and gear (also cargo)
The irony on all these numbers is they allow the driver weight of 150 lbs. as included in the numbers. I s'pose if you want to be anal about the calcs, then you need to consider the difference between your actual weight and the pseudo driver weight. I figure the truck has some level of safety factor designed into it. So, if my numbers are +/- 100 lbs, then close enough.
There are a few numbers that are important to safe loading and towing within limits of the truck:
GVWR (Gross Vehicle Weight Rating) = Maximum weight of the truck and cargo with a full fuel tank and 150 lb driver.
TW (Tare Weight) = Weight of the truck with full fuel tank and 150 lb driver. No Cargo.
CC (Cargo Capacity) = Difference between GVWR and TW; GVWR-TW = CC
Anything added to the truck beyond the full fuel tank and 150 lb driver is considered as Cargo. For instance: remove stock bumper (assume 100 lbs) install aftermarket bumper/winch combo (assume 250 lbs), TW increases by 150 lbs, CC reduces by 150 lbs, GVWR remains static. A 300 lb driver hops in, then TW increase and CC decreases by 150 lbs.
Hope this helps.