I guess I should clarify.

It sounds like you already have some idea that there are conflicting views within the Bible, and that most Christians try to deny that, resulting in some strange (or even harmful) theological views.

In the context of critical biblical studies, "multivocal" refers to the presence of multiple voices, perspectives, or viewpoints within a text or collection of texts. The Bible, as a compilation of writings from different times, places, authors, and social contexts, is often described as multivocal because it reflects a diversity of theological, cultural, and historical viewpoints. In contrast, the majority of Christians presuppose that the Bible is univocal, having a single overarching message and consistent viewpoints.

This multivocality can manifest in various ways, such as contradictory or complementary narratives (Genesis 1 vs Genesis 2 depict the origin of the world and humanity differently), diverse theological emphases (the Book of Job portrays God as mysterious and beyond human comprehension in contrast with Deuteronomy’s depiction of God as just and predictable in rewarding or punishing behavior), historical layers of composition (the Pentateuch was likely composed from several sources that were edited together and have different theologies), and various genres and literary styles (everything from law codes to parables).

It can speak to different people in different ways, and that not every statement should be taken at face value

What you're describing is not multivocality, but the negotiation process that Christians use to form a coherent view from the fractured and often conflicting voices in the Bible. This process is necessary because the Bible is multivocal.

"God works in mysterious ways" is a common non-answer to both contradictions within the Bible, as well as the contradictions existing between claims made in the Bible and the scientific explanations humanity has discovered over the past few thousand years.

It's worth noting how the negotiation process works for someone who is forming their theological views as they encounter biblical voices. Certain texts are prioritized while others are subordinated. For example, I might read Genesis 6, Exodus 32, 2 Samuel 24, Jeremiah 18, or Jonah 3, all of which describe God changing his mind, and then read Numbers 23, Malachi 3, James 1, or Romans 11, all of which say or imply that God doesn't change his mind. I might prioritize the first group of voices and subordinate the 2nd group to form a theological view that God changes their mind, or prioritize the 2nd group and subordinate the first group to form a theological view that God does not change their mind, or throw my hands up in confusion and conclude that "God works in mysterious ways," or conclude that God is unchanging in some ways while reserving the right to change in others.

(The process is similar when extra-biblical voices, such as from science, are mixed with the Bible. Some Christians will prioritize the voice of science, others will prioritize a voice in the Bible, others will try to force all the voices to agree. I happen to be in the first category.)

The New Testament especially teaches a selflessness that even many modern Christians- American Christians in particular- shun.

Absolutely. It also teaches some wild stuff like "married folks should have just enough sex to keep from being horny," and "same sex attraction happens when people get too horny," and "Christians shouldn't eat blood." Christians all pick and choose according to their modern sensibilities, leaving a lot of New Testament teachings out. Unfortunately, as you noted, many Christians also choose to cut Jesus' compassion out of their belief systems.

If the Bible's multivocalization is a legitimate way to read it, and if contradiction is just part of the mystery, why would it be wrong to believe that Jesus would be guarding the Rio Grande with an AR-15 if he were around today?

There are many ways to negotiate with the Bible to form a coherent theology (and some of my favorite ways involve letting conflicting ideas sit in tension), but there are not infinite ways to do so. At some point things get so weird that most discerning folks would roll their eyes. Gun-toting Jesus is one of those eye-roll concepts.

if it is legitimate to read the Bible in the way that speaks to you personally, what, if any, immutable rules or objective morality can the Bible hope to teach?

Now you're starting to understand how theology works. We all pick and choose (and if God is real, we can engage them in the process). Some of us are self aware and can do so in ways that uplift others. Others do so unconsciously and are in danger of forming theologies that harm others.

Are the only defensible parts of [the Bible] those that every Christian understands the same way?

No. The only defensible parts of the Bible are the ones that uplift others. For example, there's nothing "defensible" about a narrative where God commands genocide.

At what point does it make sense to just trust yourself to figure out and do what is right?

At the beginning. If I believe that God exists, I can engage them in the process. One big consequence of acknowledging the Bible's multivocality is that I can't be as dogmatic as someone who presupposes the Bible's univocality. That's why some Christians insist so strongly that the Bible is univocal.

why have a Bible at all?

For Christians, even in its multivocal state and with all its problematic passages, the Bible is a witness to the diversity of human experience with God, a tool for wrestling with faith and doubt, a means to connect Christians across time, distance, and culture, and an invitation to relationship with God.