To make it easier for you to understand the benefits of pellets over flakes, I'll break down the things into simpler composition. I might still go deep about this so please let me know which part you need further clarification.
Without putting the ingredients into perception first, let's see what makes floating pellets, sinking pellets and flakes quite different from each other.
Sinking pellets
1. The stuff does sink which does not add up air when the goldfish attempts to swallow them as fast as they sink. Advantage.
2. Pellets hold the much needed vitamins better than the flakes do. Advantage.
3. It does pollute. It should be treated as food leftover that acts as pollutant. This is more advantageous to use if your goldfish act faster, have barebottom tank or substrate is less coarser to prevent the pellets from slipping underneath out of reach from the fish. Disadvantage.
Floating pellets
1. The stuff floats and can inject air in the process when it is swallowed by the fish. Disadvantage.
2. See sinking pellets, no. 2. Advantage.
3. Pollutant if left uneaten but as it is afloat, it is much easier for the fish to locate them immediately. Presoaking will allow them to sink much slowly. Advantage.
Flakes
1. Floats and adds up air. Disadvantage.
2. Can swell quickly and will swell even if ingested. Disadvantage.
3. Vitamins are water soluble and therefore likely to leach. I compare this to eating potato chips. May be presoaked in liquid vitamins which is tedious and time consuming. Disadvantage.
4. Deteriorates quickly after a series of reopening the package exposing to air and light. Both elements can destroy the food quality. Disadvantage.
5. Easy to locate as it does not sink easily. Advantage.
Clarifications of some fallacies:
1. "Does not pollute." This implies more to nutritional analysis than the environmental cause. A majority of commercial foods pack up very low fiber while trying to focus their ad5 on proteins which they manipulate with starch-based ingredients. This is good for less pollution risk but it increases incidence of bloating and constipation as the low fiber does not allow the fish to discharge excess foods immediately.
2. "Feed within a few minutes (time recommended varies from 2-5 minutes)." This depends how fast and how much food your goldfish need to eat. In my opinion, I'd rather the fish be given what they can consume in less than a minute especially if they are fast eaters. Drop as much food that disappear quickly into their mouths as possible but do not be tempted to drop more than they can locate and eat otherwise you end up vacuuming the wasted food anyway.
Few small meals are much better than a few large meals. 2-3 times a day feeding should be sufficient enough for juveniles. Once a day for adults.
In this part, I will include other variables this time. It seems unrelated to the original questions but I just thought I'd add this up to weigh the possibilities why we need to be cautious with the foods we choose for our fancy goldfish especially if you own ones with very compact body shape.
Causes of Buoyancy Issues:
1. Floating foods. In this case, both floating pellets and flakes are involved. Flakes are already a widely unaccepted food for some fancy goldfish predisposed to buoyancy issues especially ones with exaggerated compact body shape such as ryukins and tikus pearlscales. When the fish eats them, they also add up air into their GI (gastrointestinal) tract. As their GI tract often ends up squashed up, it becomes very difficult for the fish to expel air out of their system.
2. Starch-based foods. Many commercial foods employ the use of "fillers" (starch) to manipulate the protein outcome required for juvenile fish. The starch has been responsible in dozens of cases of buoyancy issues for being the cause of gas formation inside their tract. Goldfish do not have a proper stomach to digest their foods. They rely on the bacteria lining their tract to do that job. The bacteria in turn then reproduce gas as they digest the starch.
Unrelated...
3. Physical attacks. Some fish display unusual belligerence that sometimes results in physical damage especially internally and may not be detected. I've seen a few cases of fancies attacked by their boisterous tankmates especially koi (possibly due to spawning period) and suffered buoyancy issues as a result.
4. Bacterial and viral infections.
5. Pressure changes. Not a common issue.
Conclusions in your case:
1. I'd rather recommend sinking pellets. Your goldfish can forage the bottom very well but if they are forced to adapt on floating stuff for a period of time, then their foraging technique tends to be rather poor that sinking pellets pollute your water much quicker than you think as they are used to locating their foods around the water surface.
2. If your gravel is very coarse and traps the pellets easily, consider feeding sparingly. Do not drop all the foods at once that most of it sink immediately to the substrate.
3. NLS is fine and is one of the few commercial foods with proper nutritional analysis and ingredients (depending on the size and age of your fish). The others I often recommend are Mazuri, Dainichi, Omega One and Hikari. These ones are also very good brands but some tend to be very expensive indeed. If you are buying a large pack, try to divide them into smaller portions so you will not have to reopen the package constantly that the freshness and quality of the food deteriorate needlessly. The packs should always remain vacuum-sealed.