I prefer to leave such debates to religious scholars and focus more on the practical ramifications of these religious texts. You are absolutely correct that the Bible is very open to interpretation by believers, but I would extend that to include all religious texts, to a greater or lesser extent. I’ll go further and extend it to even non-religious texts, as we have all seen how our Constitution has been misinterpreted to mean what those in power wish it to mean.
There is a strong tendency by those who hold or seek power to use authoritative texts to support their actions. They casually disregard portions of these texts which contradict their actions, and cherry-pick only those portions which support them. When necessary, they will happily adopt an interpretation of a passage that clearly strays far from the original meaning, using elaborate and convoluted logic to explain why that text doesn’t mean what it seems to mean, but rather what it “really means”.
Although the majority of contemporary mainstream Christians reject the literal application of Old Testament law and cite this supposed New Covenant to justify that rejection, those passages are still present in the Bible. They have historically been used to justify atrocities, and they are still used by more extreme sects to justify morally reprehensible beliefs. My disdain for the Christian Bible is every bit as strong as my disdain for the Quran. Yes, the Quran and the various Hadith are more explicit in the punishments that they dictate for various offenses, but I see this as a matter of degree; when interpretation is considered, there is little difference. A more important distinction, I think, is the Quran’s call to advance the spread of Islam through warfare.
Both Christianity and Islam adjure their adherents to spread their faith and to convert non-believers. The Christian Bible generally advocates a more peaceful approach to this, although it has certainly been used as a justification for expansion through warfare (the Spanish Conquistadors come to mind). By contrast, the Quran and the various Hadith openly espouse a more militant approach.
Western analysts use the terms Islamism, Militant Islam, and Political Islam to describe those movements which embrace and promote the use of force to subjugate non-believers and expand Islamic power. Political Islam does not distinguish between religious and secular authority; to their thinking, they are one and the same. The goal of Islamism is to establish a Caliphate, within which Sharia law is the ultimate authority. Because their religious texts explicitly support such a position, it is easier for Islamists than for Christians to justify militant expansion. This is not to say that Christians cannot and have not used the Bible to justify that sort of thing, but they do have to work a bit harder to interpret their scripture to support it.
The leaders and to a great extent the citizens of moderate Islamic (as opposed to Islamist) nations generally reject Islamism. Yes, this requires that they conveniently ignore significant portions of the Quran, but we’ve already discussed how readily those in power pick and choose which texts to cite. In this, nations like the UAE can be compared to Christians who conveniently ignore the more brutal portions of Leviticus.
My bottom line, I suppose, is that scholarly discussion of the legitimacy of a New Covenant is the purview of religious scholars. As an atheist, I know that those in power will use whatever portions of the Bible they wish to justify their actions, so such distinctions have little practical value.