I understand that seaplanes are usually prop planes but for cargo planes it would be beneficial in times where large cargo supply operations need to be carried out on non-carrier operations.
Because you can't land a seaplane at high seas anyway. Seaplanes can't handle too big waves.
It is said that hitting water at high speed is just like hitting concrete (at somewhat slower, but still quite high speed). The landing has to be gentle for the sea plane to handle it (about similar to landing on the land). But if there are significant waves, the aircraft touches down, bounces of the next wave and then will hit the next one at too steep angle and be damaged or destroyed.
With early float biplanes that had landing speed maybe 20 or 30 knots it could work, and they were sometimes used as observation planes launched with catapult and then landing on the water next to the launching ship. But today wherever slow speed is enough, a helicopter is more practical, and when high speed is needed, it wouldn't work.
And the Navy did try with Convair F2Y Sea Dart, and it proved impractical. It was also other parts of that design that were unsatisfactory, but nobody ever try the water skis again because they didn't show enough advantage to be worth it.
Also Russian Navy tried with Beriev A-40 Albatros, with original intent as maritime patrol aircraft, but the only version that entered production is the civilian Beriev Be-200 Altair used mainly for aerial fire-fighting.
The real answer is because there is no need.
Prior to WW2, both the US Navy and US Army had huge interest in seaplanes for exactly the reason you mentioned: large cargo supply operations. Indeed, it was not only the military who were interested. Commercial civil aviation was also hugely interested in seaplanes. The period in history that saw the success of large seaplanes was exactly at the beginning of WW2.
WW2 had a very interesting side-effect to aviation. Because aviation was so important in the war effort the war caused thousands of airfields to be constructed. After the war a lot of these airfields evolved into airports (both military and civillian). Airports killed seaplanes. The main advantage of a seaplane is being able to operate without an airport. If you have an airport you don't need seaplanes.
WW2 also taught militaries around the world the importance of combined arms and cooperation between all the different forces you own. For the US military transport operations are primarily the responsibility of the Air Force. So the reason why the Navy does not do large cargo supply operations is because that is the job of the Air Transport Command which is part of the Air Force. And the US Air Force does not want to use seaplanes to transport cargo because they have airbases all over the world.
There is practically no location on Earth near a large body of water that is not also near an airport or airfield.
In addition to Jan's answer, to recover the aircraft you need to slow down, nearly stopped. This makes the entire ship group vulnerable.
Also, an aircraft carrier can recover aircraft at full speed and have multiple launch/recovery operations running at the same time. Swapping out a flight of aircraft takes minutes.
For cargo operations this is called "replenishment at sea" and there is a well defined method to carry that out without stopping or slowing down much.