You're strawmanning "literal interpretation" away from the best representative meaning of the phrase. There are two senses to it: (a) that the Bible is true literally and on face value, and (b) the Bible is true as written within a narrow interpretation. You're saying (a) is the case. There are novices who will ignorantly assent to (a), but if you ask their pastor or a theologian trained in their church's tradition they will say (b).

Why? (a) is impossible. Nobody who actually has biblical training will assent to (a) and when they say "literal interpretation" they do not mean (a); rather, they mean (b). A really straightforward refutation of (a) is that there are literally on face value two creation stories in Genesis. If we were to believe (a), then this implies that creation happened twice. Nobody means that "creation happened twice" when they evoke a "literal interpretation." They mean the Bible is literally true and we are to use a narrow interpretation rather than extensive Modernist elaborations. This does not mean that there are not more than one viable interpretation, all narrow. Different traditions lean toward different orthodoxies but that then does not mean that they're all "cherry-picking" - they use the same narrow standard but can end up in different places. This is simply the nature of language; there's some ambiguity, especially if we're working through translations.

With any narrow standard, it will be extremely difficult to arrive at a pro-gay marriage position because these sections are very clear. But if we adopt a looser standard, see much more as metaphor, see it as particular to a historical period, then we can arrive at more Modernist positions, but this of course allows for much more "picking and choosing" in comparison to more narrow standards.

As another poster already pointed out, most Christians who evoke a "literal interpretation" do so with preference to the New Testament. I don't know what you're saying about divorce. You realize Luther/Calvin were okay with divorce under some circumstances, right? And that's because it's up for grabs based on a strict reading of the Bible? That's the same reason that pastors were allowed to marry; they alleged that Catholicism just made a bunch of rules up that are not actually found in the Bible.

TLDR: a "literal interpretation" - correctly understood - can lead to several interpretations, but these narrow interpretations vastly decrease the variability in comparison to looser, more metaphorical Modernist interpretations.