Since this discussion is, by the author's own admission, delving into the theoretical and philosophical, I believe it is just as important to answer the question practically.
Even looking into the nature of the word physics, one would find that it has a Greek origin and meaning: nature. The coining of the word physics itself is intended to convey fundamental, source-like practicality to science. Physics should, therefore, be rooted in the scientific nature of the cosmos, another ancient Greek word and concept.
Beginning with a negative proof of time as a fundamental part of nature, I take you to the kindergarten classroom. Here, in our fundamental learning facility, we observe children who need guidance and nurturing. Our first pupil comes in from "recess," a time in which the child existed in the cosmos as a wanton, carefree denizen of the "playground," which for all intents and purposes here is a place in which the pupil encounters a dirty environment. Our first pupil's skin has surface contamination, and so the instructor demands that the pupil go immediately to the washroom and clean off the dirt from his skin.
Here is where time gets really important. The first pupil, heretofore referred to as P1, as if motivated by some unnecessary wantonness, returns from the washroom in literally no time. P1 spends 0 days, 0 minutes, 0 seconds inside the washroom, declares the cleansing complete and attempts to return to the classroom. Even a rudimentary knowledge of physics would determine that P1 is in gross error. However, that is not sufficient proof. Because even if the pupil has spent 20 minutes inside the washroom, the instructor will not permit re-entry to the classroom without evidence of a physical and/or chemical process that could have cleaned P1's skin.
We can continue this proof to point at which we arrive at a practical solution, to wit, P1's skin is clean. It will become increasingly obvious in such a situation that P1 is generally supposed to use running water, vigorous motions of the body, and a chemical solvent to perform the cleansing. While there are numerous options by which to choose from, the kindergarten teacher will often need to explain and teach the child to process the skin with an aqueous chemical solvent, surfactant, vigorous motion, and a very practical element which cannot be "found" inside the washroom: time.
Because a lot of pupils believe that washing the skin is complete immediately upon contact with the soap or water. P1, and perhaps some physicists, need to understand that there is a chemical and physical process that takes place while standing at the sink, and time is a critical element in this ... and every ... physical, (Greek: natural) process.
How many physicists have not learned this simple truth about physics itself? If there is no time, then there is no process. And with no process, there is no action. And with no action, there is no practice. Impractical physics, that is, physics with no time, is not natural. If you remove time from physics, or you disallow it or ignore it, then you must be talking about the supernatural. And that is the full circle of practical physics colliding with philosophy. A philosophical physics, sans time, is one and the same as a physics with no truths and no parameters: it is timeless, without bounds, unnatural, metaphysical, and as Pupil P1 showed us: absurd, illogical, impossible, erroneous, naive, and ignorant. It might even be a bit of a dirty trick.
The only physics possible without the element of time is one outside of this cosmos that was created with it. If you leave out time, physics devolves into one of two things: the endless paradise of God, or the bottomless pit of Hell.