Effective theories like Little Higgs models or Nambu-Jona-Lasinio model are non-renormalizable and there is no problem with it, since an effective theory does not need to be renormalizable. These theories are valid until a scale $\Lambda$ (ultraviolet cutoff), after this scale an effective theory need a UV completion, a more "general" theory.
The Standard Model, as an effective theory of particle physics, needs a more general theory that addresses the phenomena beyond the Standard Model (an UV completion?). So, why should the Standard Model be renormalizable?
Answer
The short answer is that it doesn't have to be, and it probably isn't. The modern way to understanding any quantum field theory is as an effective field theory. The theory includes all renormalizable (relevant and marginal) operators, which give the largest contribution to any low energy process. When you are interested in either high precision or high energy processes, you have to systematically include non-renormalizable terms as well, which come from some more complete theory.
Back in the days when the standard model was constructed, people did not have a good appreciation of effective field theories, and thus renormalizability was imposed as a deep and not completely understood principle. This is one of the difficulties in studying QFT, it has a long history including ideas that were superseded (plenty of other examples: relativistic wave equations, second quantization, and a whole bunch of misconceptions about the meaning of renormalization). But now we know that any QFT, including the standard model, is expected to have these higher dimensional operators. By measuring their effects you get some clue what is the high energy scale in which the standard model breaks down. So far, it looks like a really high scale.
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