I certainly don't think they should be editing anything out of the statement, but I also don't reach the same conclusion. Blue commentary is mine.
From OP:
The left-wing liars at CNN have intentionally edited the video to make it look as though Trump said “absolutely” to a Muslim registry. What CNN edited out is in bold:
Reporter: Should there be a database system that tracks Muslims who are in this country?
Trump: There should be a lot of systems, beyond databases. We should have a lot of systems, and today you can do it. But right now we need to have a border, we have to have strength, we have to have a wall, and we cannot let what’s happening to this country happen any longer.
[Assuming that he intended his words to have their ordinary meaning in English, this response means he (1) agrees that there should be databases to track all muslims in the country, (2) he thinks there should be "a lot of systems, beyond databases" in addition, (3) wants strength, and (4) wants a wall. If you edit out the bolded, it makes it look like he said the ONLY thing he wanted was a database + systems beyond a database. That's inaccurate, but not inaccurate in the way that OP wants it to be. He wants databases + systems beyond databases + strength + a wall. ]
Reporter: Is that something your White House would like to implement?
Donald Trump: I would certainly implement that. Absolutely.
Again, here the editing makes it look like the only thing he would implement was databases of muslims and lots of systems beyond those databases. That is inaccurate. As discussed above, what he was actually agreeing should be implemented was databases + systems beyond databases + strength + a wall. That's just how English works.
Trump’s “absolutely” is clearly in reference to strengthening the border. Look at the whole transcript. When the NBC News reporter asks, “Is that something your White House would like to implement?,” Trump has just talked about fortifying the border and obviously believes that is what the NBC reporter is referring to.
CNN edited that out!
No. Where neither the reporter nor Trump identify one of the four things he said we should be doing, the pronoun "that" is taken to refer to the list of things he said we should be doing. Not simply the last of the things he said we should be doing.
Bob: I want a pepperoni pizza, a beer, and a napkin
Joe: Is that something you'd like in one hour?
Bob: Yes, I would.
Bob has obviously just said that in one hour, he wants a pepperoni pizza, a beer, and a napkin. Nobody would interpret this exchange as an expression of desire for a napkin and just a napkin.