I'm sorry, but this is stupid. It's not something "being taken seriously by todays physicists" and quite frankly, this article doesn't really say anything at all.
First of all, classical physics isn't exactly a "wrong" model of physics. Newton's laws are still obeyed in quantum mechanics, but as operator equations on the X and P operators in the Hilbert space. The only difference is that X and P, instead of being numbers, are non-commuting operators.
Second, lets look at the Schrodinger equation a little more closely. In it, time and space AREN'T treated on equal footing, so it doesn't simply make sense to parametrize r as a 4 vector. You have SECOND derivates of space and only FIRST derivatives of time.
Actual physics--relativistic physics--is covariant, which means that particles and fields transform under the Lorentz group. In the Klein-Gordan and Dirac equations (and Maxwell's equations), time and space ARE treated on equal footing because they can Lorentz-transform into one another. So, space and time are part of a 4 vector. But this IS mainstream physics and has been mainstream physics for 90 years or so.
I realize I've gotten way too outraged over this, but I just finished reading Chapter 28 and the top of Chapter 29 in "Harry Potter and the Methods of Rationality", and googled "Similarly, timeless formulations of quantum mechanics" to figure out what the hell you were talking about (and I have a PhD in physics. My research is in quantum chemistry/density functional theory).