It's the other way around, the Higgs field is the field introduced to break the EW symmetry. It can do this because it is particular in that it has a non-zero v

It's the other way around, the Higgs field is the field introduced to break the EW symmetry. It can do this because it is particular in that it has a non-zero vacuum potential, that in essence picks out a "direction" in EW-space at each spacetime point that becomes special. It's like the underlying EW theory has an internal space and the equations don't care about the "origin" in this space, if you made a simulation of this and moved the origin everything would look the same. But as soon as you introduce another all-encompassing field that interacts with EW that has a specific absolute value, the equations lose the symmetry - you can't move around the origin anymore as in effect you have now fixed the numbers.