the Standard Model just happens to be perturbatively renormalizable which is an advantage, as I will discuss later; non-perturbatively, one would find out that the Higgs self-interaction and/or the hypercharge $U(1)$ interaction would be getting stronger at higher energies and they would run into inconsistencies such as the Landau poles at extremely high, trans-Planckian energy scales.
But the models where the Higgs scalar is replaced by a more convoluted mechanism are not renormalizable. That's not a lethal problem because the theory may still be used as a valid effective theory. And effective theories can be non-renormalizable - they have no reason not to be.
The reason why physicists prefer renormalizable field theories is that they are more predictive. A renormalizable field theory's predictions only depend on a finite number of low-energy parameters that may be determined by a comparison with the experiments. Because with a fixed value of the low-energy parameters such as the couplings and masses, a renormalizable theory may be uniquely extrapolated to arbitrarily high scales (and it remains predictive at arbitrarily high scales), it also means that if we postulate that the new physics only occurs at some extremely high cutoff scale $\Lambda$, all effects of the new physics are suppressed by positive powers of $1/\Lambda$.
This assumption makes the life controllable and it's been true in the case of QED. However, nothing guarantees that the we "immediately" get the right description that is valid to an arbitrarily high energy scale. By studying particle physics at ever higher energy scales, we may equally well unmask just another layer of the onion that would break down at slightly higher energies and needs to be fixed by another layer.
My personal guess is that it is more likely than not that any important extra fields or couplings we identify at low energies are inherently described by a renormalizable field theory, indeed. That's because of the following reason: if we find a valid effective description at energy scale $E_1$ that happens to be non-renormalizable, it breaks down at a slightly higher energy scale $E_2$ where new physics completes it and fixes the problem. However, this scenario implies that $E_1$ and $E_2$ have to be pretty close to one another. On the other hand, they must be "far" because we only managed to uncover physics at the lower, $E_1$ energy scale.
The little Higgs models serve as a good example how this argument is avoided. They adjust things - by using several gauge groups etc. - to separate the scales $E_1$ and $E_2$ so that they only describe what's happening at $E_1$ but they may ignore what's happening at $E_2$ which fixes the problems at $E_1$. I find this trick as a form of tuning that is exactly as undesirable as the "little hierarchy problem" that was an important motivation of these models in the first place.
The history has a mixed record: QED remained essentially renormalizable. The electroweak theory may be completed, step-by-step, to a renormalizable theory (e.g. by the tree unitarity arguments). The QCD is renormalizable, too. However, it's important to mention that the weak interactions used to be described by the Fermi-Gell-Mann-Feynman four-fermion interactions which was non-renormalizable. The separation of scales $E_1$ and $E_2$ in my argument above occurs because particles such as neutrons - which beta-decay - are still much lighter than the W-bosons that were later found to underlie the four-fermion interactions. This separation guaranteed that the W-bosons were found decades after the four-fermion interaction. And this separation ultimately depends on the up- and down-quark Yukawa couplings' being much smaller than one. If the world were "really natural", such hierarchies of the couplings would become almost impossible. My argument would hold and almost all valid theories that people would uncover by raising the energy scale would be renormalizable.
General relativity is a big example on the non-renormalizable side and it will remain so because the right theory describing quantum gravity is not and cannot be a local quantum field theory according to the old definitions. As one approaches the Planck scale, the importance of non-renormalizable effective field theories clearly increases because there is no reason why they should be valid to too much higher energy scales - at the Planck scale, they're superseded by the non-field quantum theory of gravity.
All the best, LM