Experts say that #1 buck is ballistically the best.
It gives more shot in the air to get a better chance of hits and does more tissue damage over a wide area then 00, and it gives better penetration then #4.
Gus Cotey, Jr. authored an article about a study on buckshot sizes in the IWBA journal (of Dr. Martin Fackler). His summary was, "Based on the factors of number of pellets per unit of weight, capacity to achieve adequate penetration, and the volume of tissue disrupted per unit of pellet mass, the data indicate that No. 1 buckshot has the greatest potential of all buckshot sizes loaded in American 12 gauge shotshells for effective close range performance on human assailants and game of similar size."
Notice that he described #1 as having the "greatest potential." The thing that was missing and I believe is still missing is a factory loaded cartridge that brings all the necessary features to exploit the potential of #1. The market and factory ammunition seem to overwhelmingly favor 00.
Some of the key factors for effective buckshot loads for "tactical" purposes include things like:
extra hard, copper plated shot to avoid excessive deformation
shot cups and compressible wads like Federal's "Flight Control"
Shot buffers
In addition to those features, the loads should have nominal muzzle velocities around 1075 fps, not over 1200 fps for 16 pellet #1 loads. High recoil buckshot loads are very damaging to shooter performance, and the extra velocity doesn't improve effective range or terminal performance. Instead, it often deforms pellets, opens patterns, and reduces penetration.
There are many #4 buckshot loads that do not meet the 12" penetration standard that is popular. The low sectional density of the small pellets, combined with deformation of soft pellets at high velocities (like 1350 fps) result in poor penetration. If you do use #4, it's advisable to make sure the particular load you use will penetrate.
Most 00 loads will penetrate, maybe over-penetrate and there is an opportunity to "optimize" the load for better performance by reducing shot size to obtain "adequate" penetration with greater pellet count and less recoil, hence the attraction of #1 for example. The problem is you may have to do that optimizing yourself, because Federal, Winchester, and Remington, to my knowledge, have not done it, and probably will never do it since buckshot has fallen so far out of favor with law enforcement, with the 5.56mm carbine taking its place. (that's another issue to itself)