Why can't I use $\sum_{x\in\mathbb{R}} |x|$ instead of Cantor's diagonal?

Let's ignore the proof by contradiction aspect (really it's a proof by negation, which is a different sort of thing and is constructively accepted) and focus on the pure-construction part.

Cantor does the following: given a map $f:\mathbb{N}\rightarrow\mathbb{R}$, he

1. precisely defines a new object $\alpha$,

2. shows that $\alpha\in\mathbb{R}$, and

3. shows that $\alpha\not\in ran(f)$.

Step $2$ is usually glossed over in presentations, leading to its being ignored by students, but it is absolutely crucial. In particular, the "add up all the elements of $ran(f)$" approach you suggest fails at exactly this point (even if we ignore that it also probably runs afoul of point $1$). Similarly, if we try to apply the diagonal argument to the rational numbers, the issue is that the object we produce given a map $g:\mathbb{N}\rightarrow\mathbb{Q}$ will be a real number but not necessarily a rational one.

That said, note that your "add-up-everything" idea does turn into an actual result (which I'll state imprecisely since that's not really the point):

No algebraic structure which has notions of ordering and sums of finite-or-countable sets of elements, which satisfies certain basic properties, can be countable.

It's just that the real numbers do not constitute such a structure.