Let's put Mt. 16:28 back into its context, quoting along with the verse that precedes it (Mt. 16:27, 28):
For the Son of man is to come with his angels in the glory of his Father, and then he will repay every man for what he has done. Truly, I say to you there are some standing here who will not taste death before they see the son of man coming in his kingdom.
Verse 27 (in boldface) clearly refers to the end of the world and the last judgment with the Son of man coming with his angels in the glory of God the Father and repaying every man for his actions. The only way one can interpret verse 28 as referring to the beginnings of the Christian church is to ignore the context of verse 27. Christian apologist often do this by saying that verse 27 and 28 really don't have any relation to each other, which is absurd.
Furthermore, the gospels and the Pauline epistles are filled with predictions of the imminent end of the world. This is particularly true of what is variously referred to as the "little apocalypse" and the "Olivet discourse," where Jesus,after saying that the sun and moon will be darkened, the stars will fall from heaven, the powers of the earth shaken, etc., says (Mt. 24:34):
Truly, I say to you, this generation will not pass away till all these things take place.
The best fundamentalist apologists can come up with as a defense here is to assert that the word translated as "generation," genaea, was a scribal error: It should hve read genos, meaning "race" or "people." In other words, the Jewish people will not pass away before these things rake place. Agin, this is absurd. In order to accept that view we have to view the transmission of the gospels as error-prone, which flies in the face of fundamentalist assertions that it is inerrant. Also, in the context of the verse, which is one of time, "generation" - what's actually written in the text - makes more sense than "race."