Following the wikipedia link to "Time:Time as Unreal" from Glenn the Udderboat's answer I read :-
Time as "unreal"
In 5th century BC Greece, Antiphon the Sophist, in a fragment preserved from his chief work On Truth, held that: "Time is not a reality (hypostasis), but a concept (noêma) or a measure (metron)."
If I can paraphrase your hypothesis: "It seems to me that there is no such thing as time" as: "time is an abstract concept but not a material thing" then it would appear to be consistent with Antiphon's declaration.
Let me assume that they are consistent.
Then, regarding your first question: "Does this view of time work within the current framework of phsics?"
Antiphon's declaration is certainly consistent with the Galilean/Newtonian framework of classical physics. Here time in the sense of "relative time interval" is a quality used to describe intervals between events which occur in some changing system. Relative times (epochs) of events can be quantified by reference to some standard measure of chronological time (tree-rings, volcanic eruptions, moon phases, Earth rotations, etc.. Relative time intervals (durations) can be measured using standard measures of time interval (clocks). Time is a fundamental dimension (along with Mass and Length) in Newtonian physics but note that Newton's Laws are all expressed in terms of ratios rather than absolute measures. Where Time is used in Newtonian physics it is a generally used to quantify a speed, velocity or other rate of change. Or it may be used as a multiplier to express the quantity of something which has accumulated/increased/decreased over a period of time e.g. $distance = velocity * time$.
In the Galilean/Newtonian framework it is always permissible to say that two events are absolutely simultaneous or absolutely sequential (one occurring before the other). This is not the case in Einsteinian physics (see other answers to this Question).
Regarding your second question "Do physicists have an explanation/proof about time's existence?"
In the Galilean/Newtonian framework time is not a material thing but we can make measurements of changing things and pretend that these measurements are indicators of the current property of an imaginary, changing thing which we might call "chronological time". So it might be considered that there is nothing further that we need to explain or prove about the imaginary thing called time.
Regarding the so-called "Arrow of Time" which relates to the observation that certain complex phenomena (e.g. explosions) are observed to proceed in one particular "direction" and which lead some to suggest that there is a thing called time which can only evolve in one "direction" ...this can be dismissed as a spurious conclusion based on inadequate understanding of the evolution of the special complex dissipative systems which give rise to such "uni-directional" phenomena (see any textbook on Complex Systems or Chaos for elaboration).
It is interesting to observe that many physical systems in our experience do seem to act like clocks which are keeping regular time with each other. For example the periods of axial revolutions and orbits of bodies in the solar system seem very consistent over the span of human observations. However this "co-regularity" is something of an illusion because variations (cyclic, chaotic and progressive) do occur but are so small that they are not obvious without careful, detailed measurements.
Regarding Einsteinian and other frameworks, whether explanations of time are required and what those explanations might be... I defer to others more knowledgeable about those frameworks.
On a general note Physics and other sciences can be seen as basically methods of describing our environment which help us to predict and control our environment. In such a pragmatic scheme it may be useful to invent abstract concepts like Time and Space because they help us to describe our observations, make models and extract rules which we can then use for prediction and influence/control. Such concepts (Time, Space) do not have to exist as material things in order to be useful. We can do without the ideas of physical Time and Space by using qualities of "when-ness" and "where-ness" which are clearly-definable, relative, measurable properties within any material system.
Accordingly it can be argued that attempts to investigate the physical nature of Time and Space are misguided and that Physical Reality can be described essentially as "(material) stuff moving about". One might be tempted to add something poetic like "...in the ever-changing Present" but that would refer us back to the triad of past-present-future which is merely a (useful) conceptual invention.