It looks to me like you haven't gotten your head around the notion of an affine space.
In short: Utilites are not being added there. They are being linearly combined with coefficients summing to 1. I.e. you are taking an affine combination of them. Not a general linear combination, such as adding them (if you were adding two of them, the coefficients would sum to 2.)
If x and y are utilities, (x+y)/2 is meaningful, as is x/3+2y/3, as is 2x-y, etc. x+y is not meaningful, nor is 2x, or -x, or 3x-y.
Edit: To be clear, by "meaningful" here I mean meaningful as utilities, not meaningful as absolute numbers. Obviously none of these are meaningful as absolute numbers -- to accomplish that, you need something like (x-y)/|z-w|.
Utilities are elements of an (ordered) affine space -- not a vector space. Hence why (if represented by real numbers) they are only unique up to affine transformation.
I am not going to explain why, because this is bog-standard VNM. I am just hoping that by presenting a missing concept (that of an affine space) I might clear up some confusion.
You might also want to read John Baez's thing on torsors that satt linked.
It looks to me like you haven't gotten your head around the notion of an affine space.
In short: Utilites are not being added there. They are being linearly combined with coefficients summing to 1.
My goodness, this is getting ridiculous! Do I have to explain the concept of adding now? See those "plus" symbols? They represent the standard mathematical operation of addition. Utilites are being added there. For another example of adding utilities (with slightly less potentially-confusing math nearby) see the very next section of the same docum...
A common mistake people make with utility functions is taking individual utility numbers as meaningful, and performing operations such as adding them or doubling them. But utility functions are only defined up to positive affine transformation.
Talking about "utils" seems like it would encourage this sort of mistake; it makes it sound like some sort of quantity of stuff, that can be meaningfully added, scaled, etc. Now the use of a unit -- "utils" -- instead of bare real numbers does remind us that the scale we've picked is arbitrary, but it doesn't remind us that the zero we've picked is also arbitrary, and encourages such illegal operations as addition and scaling. It suggests linear, not affine.
But there is a common everyday quantity which we ordinarily measure with an affine scale, and that's temperature. Now, in fact, temperatures really do have an absolute zero (and if you make sufficient use natural units, they have an absolute scale, as well), but generally we measure temperature with scales that were invented before that fact was recognized. And so while we may have Kelvins, we have "degrees Fahrenheit" or "degrees Celsius".
If you've used these scales long enough you recognize that it is meaningless to e.g. add things measured on these scales, or to multiply them by scalars. So I think it would be a helpful cognitive reminder to say something like "degrees utility" instead of "utils", to suggest an affine scale like we use for temperature, rather than a linear scale like we use for length or time or mass.
The analogy isn't entirely perfect, because as I've mentioned above, temperature actually can be measured on a linear scale (and with sufficient use of natural units, an absolute scale); but the point is just to prompt the right style of thinking, and in everyday life we usually think of temperature as an (ordered) affine thing, like utility.
As such I recommend saying "degrees utility" instead of "utils". If there is some other familiar quantity we also tend to use an affine scale for, perhaps an analogy with that could be used instead or as well.