Yes and No
Authorship of the gospel
First, we must consider who authored the fourth gospel. Tradition attributes this gospel to John, a fisherman who became one of Jesus' first disciples. Because most working people of that era were illiterate, many modern scholars have questioned whether John could have written a gospel at all, let alone one filled with the complex, abstract theology we find in the fourth gospel (e.g. "The word became flesh," "you must be born from above," or "I am the true vine"). Some scholars have alleged this gospel was a forgery, while others have speculated that it was written by one of Jesus' Judean disciples.
But there is another possibility: John had help. In modern societies with nearly universal literacy, we define authorship much more narrowly than ancient cultures did. An author did not necessarily physically put the pen to the parchment himself. (Jeremiah didn't write down his own prophecies, and Paul rarely wrote with his own hand.) NT scholar Raymond Brown has identified five levels of authorship in ancient times. It's possible that John the former fisherman supplied the basic framework, and his disciples filled in some of the theology. I'll point to some specific evidence of this later.
Publication of the gospel
Second, we must consider not just the date of composition but the date of publication. This gospel was not quoted by any church leaders until the late second century. Whether this is because they didn't know what to make of its radical differences from the other widely-circulated gospels, or because it simply wasn't distributed until later, it's hard to say. But there was precedent for keeping a sacred book for use among a small group of disciples for a time before releasing it to the outside world. Both Isaiah and Daniel in the Hebrew Bible contain instructions to do this. The gospel of John does not, but it does show some evidence of having been written long before publication.
Now in Jerusalem by the Sheep Gate there is a pool, called in Hebrew Beth-zatha, which has five porticoes.
In general, the gospels are written in the past tense. The use of the present tense in this verse may indicate that this pool was still in use when this passage was written. If so, then at least part of this gospel was written before the fall of Jerusalem. That would put it contemporary with the synoptic gospels. But although church leaders of the very next generation were already circulating and quoting the synoptics, they made no mention of John.
It seems likely the first edition of John's testimony was written for local use only, and that the final edition--what we know as the Gospel of John--was published much later.
John gives us many more details than the synoptics on Jesus' travels through Galilee, Samaria, and Judea. However, at certain points the travel narrative doesn't match up.
In chapter 4, Jesus is returning from Passover in Jerusalem to Galilee. In Galilee he heals the son of a royal official, then (chapter 5) he immediately returns to Jerusalem for "a festival of the Jews". After performing another healing he gets into a dispute with the religious leaders. Then (chapter 6) he gets in a boat and crosses the Sea of Galilee.
It's hard to fit the pieces together in the order they appear in the gospel, but it all fits together if chapter 5 follows chapter 6 chronologically. There is evidence that the gospels writers generally did not consider chronology to be as important as we do today, but that's beyond the scope of this question.
Additionally, the healing in chapter 4 is called "the second sign that Jesus did after coming from Judea to Galilee", yet the first sign (changing water into wine) is recorded before his trip to Judea. And, of course, the cleansing of the temple--placed in Jesus' final week in the synoptics--appears very early in John's gospel.
If, as I've outlined above, John's testimony was written early and only formed into the gospel later, this apparent rearranging might have been part of the process of going from simple testimony to full-fledged gospel.
As you point out, the end of chapter 20 looks like a conclusion. If I'm correct that the gospel was based on an earlier written testimony, this may very well have been the end of that earlier work. It certainly makes for a strong conclusion with Thomas' dramatic confession.
But Chapter 21 is the conclusion of the gospel itself. It not only identifies the author of the original testimony, but confirms that testimony.
This is the disciple who is testifying to these things and has written them, and we know that his testimony is true.
The "we" here would be John's disciples who helped write the gospel. So this postscript might be considered a later addition because it was likely written years after John's testimony, but it's not a later addition to the gospel, because the gospel was not complete and ready to circulate until this chapter was written.