Since we don't know what is the origin and physical interpretation of the Higgs potential or the Mexican Hat. Is it subject to interpretation like we do in the quantum wave function? Is the temperature bouncing the higgs at the middle also just for sake of illustration.. meaning there is possibility the Higgs potential are controlled not even by temperature but another field hidden for example remnants of the inflaton field or dark energy that triggers the phase transition to produce non-zero vev and consequent electroweak symmetry breaking? Any papers in arxiv for this stuff?
In the time history of the universe, where temperature can be used as a parameter too,
we are and do our experiments and theories at "present time on the right.
All elementary particle fields are posited to cover the whole space and time , and they each have a vacuum expectation value of zero, throughout, except the Higgs field. The vacuum expectation value of the Higgs field is zero before the Weak(on the plot) symmetry breaking point (at ~100GeV) and after symmetry breaking its vacuum expectation value becomes 246GeV,( a parameter that fits the data,).The particles in the table acquire mass, but they still have vacuum expectation value zero , i.e energy is needed to get real particles, they cannot appear from the vacuum. This model fits the experimental data very well, and the recent discovery of the Higgs boson validates it.
There have been other mechanisms proposed for the Higgs mechanism. Possibly in new colliders built specifically to study Higgs production a different version will fit new data better. For now the current model is validated, and new proposals have to wait for more accurate data and maybe more Higgs bosons discovered.
In physics there is no "solid proof", except continuous validation of predictions.