Why is it said that light can travel through empty space?
I guess what a lot of people mean when they say that light travels through empty space is that it doesn't require a physical medium (matter) to travel through - like sound waves do.
The fact of the matter is that light does travel in the electromagnetic field, but the electromagnetic field is just a mathematical tool - as are all the other fields in QFT. You can't touch the EM field, just like you can't touch the fields that give rise to physical matter. You might say that we can touch matter and so those fields are more real. However, the sensation of touching those fields is just a result of electromagnetic interactions between the particles in ourselves and those in the object we are touching, and the reason for that being true is no more real than the reason that light can move from A to B.
To answer your specific points, the Michelson-Morley experiment proved that light travels at a constant speed for all observers. It proved that there was no physically-measurable medium in which light travels through (like sound). The electromagnetic field is how we describe light moving, however, it is not a physical object - it is just a mathematical tool and so only exists on paper.
I think the confusion in terminology comes from people trying to attach physical meaning to fields. Maybe you can attach meaning to them, and one day we will be able to look at them in a different way to how we can now. However, it must be noted that the fields introduced in QFT are mathematical tools, and that is all they are. We should not imagine them as filling space like a fluid, they simply make useful and accurate predictions.
In summary, they tell us how everything works - they will never tell us why.
Why do some physicists say light can travel through empty space?
What a physicist means by this is that there exist non-trivial vacuum solutions to Maxwell's equations. A vacuum solution is one where the sources are zero. In the case of Maxwell's equations that means that the charge density and the current density are zero. Additionally, the specific solutions are also solutions to the wave equation and waves are often described as "traveling".
Of course, "there exist non-trivial vacuum solutions to Maxwell's equations which are also solutions to the wave equation" is not a description that will really help most people. So instead we say "light can travel through empty space". This indeed conveys the idea well, with an unavoidable loss of technical precision. It is a fine non-technical statement, to be understood without too much nit-picking.
saying light is the field. Frankly, that sounds like doublespeak.
I think that is actually the correct statement. The idea that light travels through the electromagnetic field is strange. If I say that a plane travels through the air that means that there is some distinction between the plane and the air. They are different things and one is physically located in the other. That does not correctly express the relationship between light and the electromagnetic field.
Light is a specific type of solution to Maxwell's equations, which defines electromagnetic fields. Specifically light is a vacuum solution with travelling waves (perhaps even specifically waves with wavelengths in the optical part of the spectrum). As such, light is indeed an electromagnetic field and there is no in-principle distinction between it and the field. So saying that light travels through the electromagnetic field is more like saying that wind travels through air than that a plane travels through air. You can say it, and you are not completely wrong, but it is strange.
When it is said that light can travel through empty space, it is meant that it does not need matter to propagate.