The line of continuity from Hebrew text to Christian gospel was intended by the gospel writers, but I have my doubts about its necessity to Christian truth. If we take Matthew's quotes as an example, he writes a lot that things "took place to fulfill what was spoken through the Lord through the prophet." It gives the impression that these were fixed points in time, unable to be changed because a prophet had identified them as a sign, and therefore an indication that what happened was divinely appointed and therefore true.
If that's the case, what should we do with all the prophetic verses which haven't yet been actualised? I think we have two options open to us. We can either wait for them to happen (this life or the next, it seems) or we can look at the relation between prophetic texts and Christ in a different way.
I suggest that rather than saying the prophetic texts legitimise Jesus as the Christ, we should say that Jesus legitimises texts as prophetic. A verse, or a quote, or a symbolic picture is only given value through the interpretive lens of Jesus, and not the other way around. Furthermore, this doesn't mean that Jesus legitimises the text so that we can say that the text was right all along. I think it means that Jesus has invoked the text, much like we would quote someone today, in order to bring some implicit baggage with it.
For the gospel writers who quote scriptures, I think we are left with a situation in which they want to associate Jesus with a thread of Hebrew thought. It's like saying, "Jesus reminds me of the book of Isaiah's this way." Quoting a small piece of text from one Jew (Matthew) to another (his readers) is like two nerds quoting lines from The Simpsons or Monty Python to each other. Just one line is enough to trigger the whole scene in the mind of the audience.
I'm going to leave you with the mental image of a group of nerdy rabbis quoting lines from Ezekiel and erupting in a chorus of, "Ah hah!"