...I wasn't going to reply to this -- I can't expect to quickly correct the macro-mistake you're making -- but in this case the micro-mistake you're making is a very simple one, so I can point it out.
Yes, the plus sign represents the operation of addition. But it isn't utilities being added; pay attention to what the summands are! The summands are not utilities, but probabilities multiplied by utilities. At no point in anything you have pointed to are two utilities added without first being multiplied by probabilities (because that would be a meaningless operation).
Now, it's true that if x and why are utilities, then x+y is the same as 1x+1y, and so this is a special case of "multiplying by probabilities and then summing". But in fact the general operation of "multiply by probabilities and sum" is not meaningful; it is meaningful only in the special case when the probabilities in question sum to 1. (Though as I said above, it's slightly more general than that, in that they don't have to be probabilities -- they can be any real number.)
Every "sum of utilities" you've pointed me to -- which, as I've said, have not been sums of utilities, but rather sums of utilities scaled by real numbers -- has taken this restricted form. Which is good, because otherwise the reasult would be meaningless. (Well, meaningless as a utility, anyway -- we could consider something like x/3 + y/3 + z/3, where x, y, and z are utilities. Then you could point out that this contains the subsum x/3+y/3, which is not of the right form. But while meaningless as a utility, it's a perfectly valid 2/3-of-a-utility. You could mulitply it by 3/2, or add another third-of-a-utility, or or have it and then add a 2/3-of-a-utility, and get a meaningful utility.)
(If I really wanted to pick nits, I could point out that "+" only really stands for addition if we first embed the affine space of utilities in a vector space, or assign particular real numbers to utilities; otherwise it's just a notational shorthand. But in the case that we do assign real numbers to utilities, yes, it's standard addition. Just not utilities that are being added.)
Yes, the plus sign represents the operation of addition. But it isn't utilities being added; pay attention to what the summands are! The summands are not utilities, but probabilities multiplied by utilities.
Your mistake is even simpler. Probabilities are unit-free. They are numbers between 0 and 1. As such, they are dimensionless. So: a utility multiplied by a probability is still a utility.
If it helps any, I have a degree in mathematics. I do actually know what I am talking about here.
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.