The method is marvelously clever and straightforward and I can see people thinking it's very cool (which it is).
I do feel it would be clearer if it began like this: "Let $x$ and $y$ be the two numbers. Since they sum to $-7$, their average is $-7/2$ and so one is more than $-7/2$ and the other is less. Write $x=-7/2 + d$ and $y = -7/2 - d$." Then you're explicitly declaring that your real goal is the two numbers and this other number is just a means to an end.
The idea that if two numbers add to $A$, then one must be a bit more than $A/2$ and the other must be a bit less than $A/2$ ... I feel this idea would be quite a conceptual leap for many students. It's certainly something that it is worth them knowing, but I can bet quite a few of them will be very confused by that particular step.
There's a little nuance in there that the number $x$ you have chosen is implicitly positive. Students familiar with putting a $\pm$ when they perform a square root will get two answers for $x$ and sub them both back in to get four answers (which turn out to only be two answers after all). This will waste quite a bit of time unless you explicitly point out that the $x$ you're using is positive.
The calculations involved are actually exactly the same as those you do when completing the square: divide the x-coefficient by 2, square it, add this to the constant, square root, add and subtract this to minus half the x-coefficient. So in terms of calculations it is actually equivalent to completing the square. Completing the square feels harder for some reason to me -- I'm not sure why. Still, completing the square is a method they probably need to know for general quadratics, so is it worth showing them something else for a particular problem?
The major problem to me is that the reasoning in it only applies to this specific sort of problem, and so might encourage the "list of types of problems - list of solution methods" approach to maths. I can't see any other context in which this type of reasoning would come up, so it seems to me that you'll be teaching them to say to themselves "there's a clever trick to these ones, let's see if I can remember it" rather than "If I just use plain ordinary reasoning and common general methods I'm sure I can figure it out."
Finally, there are actually thought processes that will tell you precisely whether the factors have to be positive or negative (and you always add them since it says "sum"). If the product is positive then it must be that both factors are positive or both are negative. Then if the sum is positive this is only possible if both factors are positive, and if the sum is negative this is only possible if both factors are negative. On the other hand if the product is negative then it must be that one factor is negative and one is positive. Then if the sum is negative it must be that the negative factor is bigger in size than the positive one, and if the sum is positive then it must be that the negative factor is smaller in size than the positive one. So in short:
SUM + SUM -
PRODUCT + both factors + both factors -
PRODUCT - big factor +, small factor - big factor -, small factor +
In your example, the pairs of factors of 60 are:
1, 60; 2, 30; 3,20; 4,15; 5,12; 6,10.
Since the product is negative we have to have one + and one -. Since the sum is negative the big one has to be -ve. So the pairs of factors we need are actually:
1, -60; 2, -30; 3, -20; 4, -15; 5, -12; 6, -10.
So it must be 5 and -12.