There are many examples of the preference axioms of consumer theory being violated. I feel like there are very few cases where at least one of these axioms isn't broken, so is any theory or model created under these assumptions representative of the real world?
The violation of the preference axioms is frequent in the real world, which questions the usefulness of neoclassical model results. This is something that most neoclassical economists will say that it is just "noise", or has a minimum effect, while there is limited empirical data to sustain this - maybe some lab work (controlled experiences), but not real world observations that I know of. The consensus in most policy making institutions and economic universities it is, still (and unfortunately), that neoclassical assumptions are what should be used.
You can read Transaction cost economics for a different perspective which doesn't assume the preference axioms, from Oliver Williamson, Nobel Prize in 2009. It refutes the preference axioms, stating that there is bounded rationality in every choice: humans have limited reasoning and searching capabilities.
Here's a summary of his work on TCE:
Williamson’s contributions to the field of Transaction Cost Economics complement, and extend, those of Coase. First, Williamson started with an explicitly behavioral assumption of human behavior (bounded rationality). Second, he recognized that transacting parties sometimes behave opportunistically and take advantage of their counterparties. Finally, he identified features of transactions (e.g., specificity, uncertainty, frequency) that cause markets to fail; and hence, are likely to lead certain transactions to be organized within firms (hierarchies) rather than markets.