But, if the Standard Model is treated as an Effective Field Theory that emerges at low energy, shouldn't the Electroweak Theory be unnecessary?
No, for 4 reasons.
It would be nice to combine the theories together, why have 2 seperate ones, when under the right conditions, you can explain all of the force carriers involved in 1 combined theory.
The electroweak theory has led to the concept of the photon as being viewed as a combination of the $B^0$ and $W^0$ field quantums.
Although we can treat them as seperate at low energies here on Earth, if they were combined at the Big Bang era, then they should be treated as one theory at high energies and may lead to a unified theory (eventually).
This reason is of historical interest today, but the quest to join the two theories together using the relatively new symmetry breaking ideas, eventually lead to the development of the Higg's Mechanism.
Please forgive a small historical piece, as this period of time was vital in the development of the Standard Model (and the Higg's mechanism) which are obviously an intergral part of physics today.
To go back in time 60 years, there was a dichotomy, the force carriers of weak of weak interactions violate parity, whereas the electromagnetic force carriers do not. This partity violition implies that weak interactions only couple to left handed particles, not to right handed ones. This was experimentally verified by Madams Wu's experiments.
Electromagnetism preserves parity, while the weak interactions violate it.
So how do you overcome this problem? You can introduce two different symmetries, one that treats left and right handed particles the same and one that does not. The trick here is to combine these symmetries in such in way that the overall symmetry is unbroken. This was done by Glashow in the 1950's.
The usual example given to illustrate this idea is of two gear wheels, each of which can independently rotate, that's the original two symmetries, but when you join them together, you lose a degree of freedom, as they must rotate as a unit.
This did not lead to a renormalisable theory of electroweak unification, but it did lead to the prediction of a new particle, the Z particle, but because nobody had actually found such a particle, or at least it identified as such, this new particle prediction was not given much notice.
This breaking of symmetries was done by hand, there was no mechanism given as to how it would happen, it was performed just to see would anything result from it.
In 1964 Salam and Ward came up with the same idea, but predicted a massles photon along with three massive weak gauge bosons. Again, they did not provide a mechanism for spontaneous symmetery breaking, it was an arbitrary decision.
Around 1967 Weinberg was involved in the investigation of spontanious symmmetery breaking, and eventually arrived at a set of symmetries that predicted a massless, neutral guage boson, which initially he wanted to get rid of, as it fulfilled no known function.
It later dawned on Weinberg that what he has discovered was a set of symmetries that produced the electroweak force and he was able to make a prediction for the masses of the W and Z bosons, which because the earlier symmetries were broken by hand, the researchersassociated with them could not do.
Significantly, in his paper "A Model of Leptons", Weinberg suggested this new electroweak could be renormalisable.
Weinberg's theory explained why all the fermions acquired mass, as well the the gauge bosons.
In 1971, t'Hooft proved that that, even with massive gauge bosons, spontaneous broken gauge symmetries, as Weinberg's was, are renormalisable.