Measures: Sigma-Additivity Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome!
What reasons are there to restrict measures to countable additivity rather than uncountable additivity? Is there a deeper reason than just it "works"? Any scratchy ideas are welcome...
 A: Uncountable additivity are considered especially in set theory. 
For example, an uncountable cardinal $\kappa$ is a real-valued measurable cardinal if and only if there exists a nontrivial $\kappa$-additive probability measure on $\kappa$.
Although in some cases $\sigma$-additivity is enough. For example, the least cardinal which has a $\sigma$-additive probability measure is alway a real-valued measurable cardinal.
However, it should be noted that the existence of such a cardinal can not be proved in the usual foundation axiom of mathematics, $\mathsf{ZFC}$. Moreover, the statement that such a cardinal exists has very high consistency strength.
A: Mainly, sets and numbers rely on different footings.
That brings structural similarities but also structural discrepancies.
These allow one to study homomorphism as are measures only to a certain extend.

To start with, both give rise to an algebraic structure by a binary operation admitting a neutral element:
$$A\cup\varnothing=A=\varnothing\cup A$$
$$x+0=x=0+x$$
Now, these extend to finite operations due to associativity:
$$A_1\cup\ldots\cup A_N$$
$$x_1+\ldots+x_N$$
(Also noting that commutativity holds for both!)
At this point finite additivity can be studied...
Next, the union immediately generalizes after a reformulation:
$$A_1\cup\ldots\cup A_N=\{x\in X:x\in A_1\vee\ldots\vee x\in A_N\}=\{x\in X:\exists i\in\{1,\ldots,N\}:x\in A_i\}$$
arriving at the union over arbitrary collections:
$$\bigcup_{i\in I}A_i:=\{x\in X:\exists i\in I:x\in A_i\}$$
The sum however requires the introduction of a Hausdorff topology.
After this first lack arriving at the sum over in principal arbitrary collections as well:
$$\sum_{i\in I}x_i:=\lim_{J\subset I:\#J<\infty}\sum_{j\in J}x_{j}\quad J\leq J':\Leftrightarrow J\subseteq J'$$
(Note that both extensions are independent of reordering by construction!)
The main two differences however are that not every arbitrary collection of numbers has a convergent sum and if so then only countably many numbers can be nontrivial:
$$\sum_{i\in I}x_i\text{ exists}\implies x_i\neq 0\text{ countably many}$$
The first point forces positive measures to lie within the extended positive real axis and complex measures within a bounded disk:
$$\mu(A)\in\overline{\mathbb{R}}_+\text{ resp. }\mu(A)\in\mathbb{D}\subseteq\mathbb{C}$$
The second point is that tells us* (but doesn't force us**) to better consider merely countable collections.
*A bad situation usually mentioned: $\sum_{x\in[0,1]}\lambda(\{x\})=0\neq 1=\lambda(\bigcup_{x\in[0,1]}\{x\})$
**In fact, the Dirac measure possesses the uncountable additivity!

So sigma additivity will be the most prominent candidate for structure preserving homomorphisms a.k.a. measures.
