Umm, no. There is a huge selection bias at work here. Most people, when they read the bible, read Genesis and Exodus, the Gospels, Psalms, Ecclesiastes and a couple of other books. It is true that lots of phrases, images, stories, quotations and so on from those heavy-hitting parts of the Bible have made their way into the general culture.
The vast majority of the Bible is not at all influential on either our religious understanding or on general culture. Partly because most of it just isn't that commonly read. The popular parts are read thousands of times more commonly than the least popular parts, and many of the readers of the least popular parts (in our era) are people who have a deep religious commitment to reading the Bible, but who aren't necessarily prolific writers or influential thinkers.
The other fact at play here is that most of the unread parts are unread for a reason. There is a lot of material which is, of course, of historical or theological interest to experts, and which had undeniable spiritual significance to many people who lived long ago, but which is very hard for most modern readers to extract any real meaning or value from.
Some examples - each of these is taken not from the really obscure books, but from 'second-tier books', in order to demonstrate how quickly the long tail of Bible literature descends to filler.
The book of Numbers (part of the Pentateuch/Torah) mainly consists of pages and pages listing how many animals of different types should be sacrificed in various ways for different festivals. This is extremely repetitive stuff, pretty much beyond satire. There is no justification given for any of this doctrine, and once you have extracted a minimal amount of historical content ("it seems like cattle were more valuable than sheep!") it's on a par with reading the telephone book.
The book of Proverbs sounds fun, but do you know any of the proverbs it contains? Probably not many, because they consist of dozens of very small variations on the same message to embrace wisdom, as though there were some doubt about the advantages of wisdom. Sample: "Blessings are on the head of the righteous, but the mouth of the wicked conceals violence." The catchy phrasing and apt metaphors which you would hope for simply aren't here (barring a very small number, which stand out as being slightly less banal: "iron sharpens iron").
A couple of the Epistles of St Paul are deeply uplifting and moving pronouncements of the essence of belief in Christ. The rest (circa 20 books) are probably fascinating political history, if you have the background knowledge about the different early Christian thinkers and communities and the evolution of their doctrine. If you don't, they are hard to make much sense of - for example, thirty pages of careful analytic debate about circumcision, responding to someone else's letter which itself breaks down a third person's arguments, neither of which we have available to us.
The awkward truth about the notion of the 'Bible as literature' or the supposed cultural importance of the Bible, is that in modern secular times, it is a myth passed on from otherwise educated people who have read less than 5% of the Bible, to other educated people who are going to read even less of it.