Regarding Hebrews 13:13, why say that “outside the camp” means “outside the rest of Christendom?” The context of the entire book and the immediate context seems to be leaving the Mosaic/Judaistic system of worship.
I agree that this seems like an “off-the-wall” reference. However, I think I can explain why some speakers use this analogy.
It would be good to start with the general principle of separation from evil. This is a complex topic because there are various forms of evil and corresponding degrees of separation. In addition, because we are in the world there is some evil we cannot separate from. 1 Corinthians 5 is an important chapter because it speaks of separation from evil in the assembly and also points out that we cannot separate from all evil (1 Cor. 5:9-10). A person may even feel free to meet with Christians at work for a Bible study who hold various beliefs without being identified as one who holds erroneous beliefs himself. We should always use every opportunity to be of help to others so long as we can avoid being identified with their errors.
Separation is not only a New Testament concept. In fact, it goes all the way back to the first day of creation (Gen. 1:3-5). There God separated light from darkness. This is the primal representation of the principle we have in 2 Corinthians 6:14 where the apostle Paul asks: “what fellowship has light with darkness?” Israel was to be separate from the nations around them (Lev. 19:2, 20:7, 26). Their failure to maintain their unique witness for Jehovah eventually lead to their captivity. Then, although some of them returned for the advent of their Messiah, they rejected the Lord Jesus. Their unbelief confirmed their position outside God’s blessing (Jehovah had declared His people lo ammi meaning “not my people”, Hos 1:9). Yet, in the early Church period there were still Jewish Christians who continued to worship at the temple even though the Lord has said “your house is left unto you desolate”. (Mt 23:38) The book of Hebrews was written to these. The apostle calls them to “go to Him (Jesus) outside the camp.” The “camp” here clearly means the worship of Judaism, as your question suggests. They were being called to join the Christian company.
How does the church correspond to Judaism? First, we need to realize in this context “the church” refers not to the true church or Assembly, but to the professed church, or “Christendom.” This distinction is based on the observation from the epistles to Timothy. In 1 Timothy we have in view the “church of the living God” (1 Tim. 3:15) but in 2 Timothy we have in view “the great house” (2 Tim. 2:20). It is in this same chapter that we are enjoined to separate from those in the “great house” and “pursue righteousness, faith, love, peace, with those that call upon the Lord out of a pure heart.” (vs 22)
So, the parallel is between “Christendom” which consists of true believers and mere professing Christians, and Judaism which consists of true followers of the Messiah and Jews many of whom were complicit in crucifying the Lord. In this analogy, “the camp” represents professing Christians which mostly do not follow sound doctrine. It is solemn indeed that the Christian profession, which started out so well has degenerated into what it is today. But, above all, it must be remembered that the admonition is to “go forth to Him (Jesus)”. We are not to follow anyone else.
A quotation from another which may help explain this point:
The atmosphere of what is called a “Christian world” [or, professing Christianity] is more detrimental to the development of faith than the unbelieving world. That is because a Christian world promotes compromise, and compromise has in it disloyalty to Christ and to His word. In the Christian world, you are not faced with what is openly hostile to Christ. Open hostility would put you on your guard, and rouse up in you all your power of resistance. Instead, you find truth owned, but not taken very seriously. You are to follow it, but not be too extreme about it. And then, we naturally grow up to the stature, morally or spiritually, of the company we keep!
Thus, the application of “go forth unto him without the camp” is the admonition in 2 Timothy 2:22, “pursue righteousness, faith, love, and peace, along with those that call on the Lord from a pure heart” instead of remaining in the great house with dishonorable vessels.
 F. W. Grant, The Numerical Bible: The Gospels, (New Jersey: Loizeaux Brothers, 1977) 104. (Edited)