While small white birds certainly are not common, it is not true that they do not exist. I  refer neither to white nor to albino genetic variants, such as white European starlings, but to such species as Rothschild's mynah, which is almost all white, apart from a blue streak over the eye. Some species of fairy tern are white, and they are not much larger than sparrows.

Exactly why any birds, including large ones like egrets and Sacred Ibis should be mainly white, is not clear to me, unless in the case of waders such as storks and great White Egret, it might in effect be a defence against overheating in strong sunlight.

The fact that small birds might evolve conspicuous colouration, need not mean that they thereby become more vulnerable to the most relevant predators. In fact vividly coloured hummingbirds and sunbirds seem to be so coloured, partly as a warning that they do not taste very good, and partly because they are almost impossible for most predators to catch. This would seem consistent with the lifestyle of fairy terns, but I know very little about the habits of Rothschild's mynahs.