No, I do not think my position is at all similar to Alex's.
I am a firm believer in the old adage "there is more than one way to skin a cat". In fact, I have always used that principle to assure myself that an answer I have arrived at has a strong probability of being correct. What I am trying to say is that, if the answer to a question requires a very specific model of the problem, my faith in the answer will be minimal. If, however, I arrive at the same answer no matter how I model the problem I then conclude the answer is most probably correct: i.e., if you get the same answer no matter how you look at the problem you can usually trust the answer. My difficulty with most scientists is that they seem to think their way of looking at the problem is the only way it can be seen. In my humble opinion, that is an intellectually dangerous tack. It is in fact, what I call a religious attack and I am of the opinion that a religious attack is intellectually bankrupt.
Because of my long time habit of looking at things from alternate perspectives, I sometimes see things that others miss. I feel the issue of time is a particular example of this.
In the interest of making relativity (in particular, the definition of time) clear, let me propose a rather simple thought experiment. Suppose for the fun of it someone has given me a functioning time machine. The time machine looks exactly like a wrist watch. To make it work, all I have to do is turn the stem (change the time indicated on the dial) and I will find myself at the time indicated. (So I won't stand still in time, if I am not turning the stem the dial advances at the standard rate if I am not moving it.) The time machine only affects the person wearing it. Also, there is a ratchet on the stem so that it cannot be turned backwards.
Now let us presume this time machine works fine (exactly as described above) and I demonstrate it to you by moving some distance into the future. Exactly what will you see? I propose that the following description of the phenomena from your perspective is exactly in accordance with the above description of the functioning of the time machine given above.
I say to you, "now I am going to move 5 minutes into the future." I then reach over and take hold of the stem of my wrist watch. I appear to rotate the stem very slowly for five minutes (such that the dial reads the correct time during that period) and then, after that five minutes have passed, I let go of the stem and say, "now do you believe it works?"
If the time machine does indeed work as described, then, as I turn the dial, I must pass through every time between the start point and the end point so from your perspective, I exist at every time between start and the end.
The purpose of the above was to prepare you for the problem of analyzing the appearances to an observer of the following situation. I will presume the old Newtonian view of the universe is absolutely correct. (If you let me hypothesize a time machine, you ought to allow me to hypothesize a Euclidean/Newtonian universe!)
First, let me make a bunch of invisible time machines. Let me include with each time machine a little computer which can tell exactly where the time machine is in that Euclidean/Newtonian universe. The time machines will work exactly as described above except that, if the time machine is moved, the little computer will advance the dial by an amount proportional to the distance moved. The proportionality will be such that if the time machine is moved 186,000 miles, the dial will be advanced one second: i.e., the object to which the time machine is attached will move an additional one second into the future if it is moved 186,000 miles (approximately one billionth of a second if it is moved one foot).
Now suppose every object in your laboratory has such an invisible time machine attached to it. What are the consequences of that? Exactly how will the behavior of those items appear to change from what you might expect? First, a billionth of a second is awful small so most experiments will not detect any effect. Second, the length of time it would take to move something 186,000 miles is so long that seeing a consequence of that one second over that kind of period would require clocks more accurate than we normally have available to us.
However, if we manage to move something a reasonable distance in an extremely short time, we might be able to see some consequences. Suppose I have a rocket ship powerful enough to get to Alpha Centauri and back in five minutes (you have already allowed me time machines and a Euclidean/Newtonian universe, the ship isn't that much more of a stretch). Now the time machine will have a significant effect! Because the ship has moved about eight light years, the time machine will move it eight years plus about five minutes into the future. Hey, guess what, although its actual velocity (remember this is a Euclidean/Newtonian universe) was greater than 500 trillion miles per hour (over a thousand times the speed of light) it will appear to be traveling just slightly under the speed of light (that 5 minutes lag) because its motion forced it into the future. Oh, by the way, it's momentum will be quite large too, certainly not mv if I use the "apparent" velocity!
Now this appears interesting. No matter how hard I try, nothing in that laboratory can "appear" to exceed the speed of light and, when I move things quickly, they appear to have more mass than they do when they are moving slowly.
As a matter of fact, it takes only a slight change in the model above to produce exactly (in every detail) the phenomena commonly referred to as "relativity". If we presume we live in a Euclidean/Newtonian universe (instead of Einstein's Minkowski space) and further hypothesize that we do not move into the future because "time has passed" but rather because we have changed position! Then all of the standard relativistic phenomena appear as if by magic. The slight change mentioned above is required to explain why we go into the future when we are not moving.
That change is very simple. All it requires is the idea that, when we think we are not moving, we are in fact moving at a high rate of speed in a fourth dimension of which we are unaware. Why are we not aware of this fourth dimension? Because everything of interest to us is also moving at the same speed and in the same direction (for the most part). Furthermore, if the momentum in that direction of everything of interest to us is quantized, then the Heisenberg uncertainty principle says that the position in that direction is unknowable (no wonder we are not aware of it). That quantized momentum will obey all the rules of mass and all of the other appearances of standard relativity become rather obvious.
It is actually not very difficult to show that the two rather different pictures of the universe produce identical experimental results. No more than a little common sense and some math are required.
It is interesting to look at the Pythagorean theorem in this strange Euclidean/Newtonian universe: (if everything is always moving at a fixed high velocity v, ds along its path is vdt)
vdt = sqrt( (dx)^2+(dy)^2+(dz)^2+(dw)^2 )
or, rearranging terms,
dw = sqrt( (dx)^2+(dy)^2 +(dz)^2-(vdt)^2 )
It seems to be no stretch at all to set that high velocity v equal to c and conclude that Einstein's invariant interval is actually analogous to the undetectable dimension above. At that point, the issue that clocks (wristwatch time) measure Einstein's invariant interval along the path of the clock -- change in w above. Leads to my comment that clocks do not measure time!
As I say, it is no more than another way to skin a cat.
Have fun -- Dick