Start with verse 21:
21 “Not everyone who says to me, ‘Lord, Lord,’ will enter the kingdom of heaven, but only the one who does the will of my Father who is in heaven. 22 Many will say to me on that day, ‘Lord, Lord, did we not prophesy in your name and in your name drive out demons and in your name perform many miracles?’ 23 Then I will tell them plainly, ‘I never knew you. Away from me, you evildoers!’
I still don't see Jesus saying directly, "Not every claimed Christian will be saved." I see what appears to be Jesus addressing those who:
1) Say to Him, "Lord, Lord," or call him "Lord" arguably.
2) It is those people who are saying they did such and such in His name.
If they're operating in Christ's name, as they claimed to be, then they're claiming to be Christians.
That time and place was not much different from operating in a Muslim country today--if a person claims to be operating "in the name of Jesus," then he's claiming to be committed.
You made a good point I think, however I don't see it as synonymous that a person claiming to operate in Jesus' name, etc ... is at the same time claiming to be a Christian. That's essentially saying that anyone who operates "in Jesus' name" is claiming to be a Christian ... what if they say they are not Christian but continue to do so regardless ? Also, at the time Christ said that, the term Christian wasn't in use, right ?
Many years ago in my teenage days, I took a date to a local psychic for fun. This psychic claimed to believe in Jesus and do things in Jesus' name, etc. She did the tarot cards, told me she got her gift from either God appearing to her or Jesus appearing to her (I can't remember exactly), but IIRC she also claimed to be a Wiccan or something to that effect. I know she probably covered the spectrum in order to market to as many clients as possible, however she was in business for many years (where her reading house was, was easily seen from a major highway in a trendy part of town so it was kind of a place you would see frequently). Now, whether or not she specifically also called herself a Christian, I don't know ... but my point is that claiming to operate in Jesus' name isn't necessarily one and the same as claiming to be a Christian. I just read a post in another thread where a man (who uses the icon, etc) claimed he is not part of the Christian religion: he is a follower of Jesus Christ, and God has been speaking to him and giving him an idea to start a new church, etc.
I realize that through the eyes of the way we use the term "Christian" today that it is basically saying the same thing, however that's because of the way we use the term today. In the Southern US, everything that comes in a soda can is a "Coke". "You want a Coke ?" "Yeah sure, what kind you want ?" "I'll take a Sprite." But Sprite is not Coke lol. It's vastly different from Coke. Similarly, I don't know about others, but I've always called tissues "Kleenex". "I need a Kleenex, can you hand me one ?" I don't call them tissue papers, or tissues, etc. But Kleenex is a brand that makes tissues. Just because it may have become synonymous with "tissue" to me, doesn't mean they are one and the same. Kleenex is one type, not the originator of tissues. LEGO would be another example. I've been corrected before on calling some off brand "LEGOs". They aren't LEGOs. I've been corrected on using the term "LEGOs" before even: interesting fact, did you know that LEGO encourages others to use the term "LEGO bricks" when talking about LEGO in plural ? IIRC it's actually company policy in official documents, etc, to not use the term "LEGOs". They don't like the term "LEGOs". I've read before that one of the reasons is because they want to make sure that their brand isn't diluted, to where when a person sees a bunch of LEGO and off brand, they just call them all "Legos" and group them together. LEGO bricks is what they want to encourage the use of.
In my mind, the term Christian came later ... like Coke, Kleenex, etc. It doesn't have a monopoly on all who claim to operate or know Jesus, yes ? And I would think, similar to LEGO (not the best analogy, but still), that the person of Jesus would have the ultimate say in what He would want His followers to be called or not called, and I don't remember seeing him endorse the use of the term "Christian" specifically, or address what a real one would be or not specifically, etc. Nor do I see where a person claiming to operate or know Jesus, is necessarily claiming to be "Christian" in the same breath.