Invariant states are not the only meaningful ones. Even in classical mechanics, a baseball traveling 90 mph toward my head is quite meaningful to me, even though it is of no consequence to my fellow mathematician a mile away.
The focus on invariant subspaces comes not from an assumption, but from the way physicists do their work. They want to predict behavior by making calculations. They want to find laws that are universal. They want equations and calculation rules that will be invariant under a change of observers.
Any particular calculation might require a choice of coordinates, but the rules must be invariant under that choice. Once we're talking about one particular baseball trajectory, that trajectory will look different in different coordinate systems; the rules governing baseball flight, however, must look the same in all equivalent coordinate systems.
The natural features of baseballs arise from the equivalence classes of trajectories of baseballs -- equivalence under the group action. Here, if we pretend the earth is flat, gravity is vertical, and air does not resist the baseball, the group is generated by translations and rotations of the plane. Any physically natural, intrinsic property of the baseball itself (such as its mass) or its trajectory (such as the speed of the baseball) must be invariant under the group action. If you don't know a priori what these properties will be, a good way to find them is to pass from individual instances (the baseball heading toward me at 90mph) to the equivalence class generated by individual instances under the group action (the set of all conceivable baseballs traveling at 90mph). Note that the equivalence class is invariant under the group action, and it is exactly this invariance that makes the equivalence class a useful object of the physicists' study.
More generally, if you are studying a physical system with symmetry, it's a good bet that the invariant objects will lead to physically relevant, important quantities. It's more a philosophy than an axiom, but it has worked for centuries.
Probably the version of "semi-simplicity" relevant in the context in those notes refers to a repn of $G$ that is admissible with respect to a compact subgroup $K$, in the sense that it is assumed to decompose with finite multiplicities over $K$. This assumption holds for repns induced from cuspidal repns on the compact itself, for example, as observed by Jacquet already in 1971. In such a context, the otherwise-too-glib remarks about complete-reducibility still reach a correct conclusion, even without completeness in a Hilbert-space sense.
The more down-to-earth situations, like $L^2(\mathbb R)$, or subspaces generated by rough functions on the circle, do not satisfy suitable "admissibility" conditions, perhaps-oddly.
Best Answer
You are misunderstanding Sternberg's argument. The argument goes like this (with my notations):
Consider two representations $V$ and $W$ of $G$ with the same character.
For any representation $S$ of $G$, let $\chi_S$ denote the character of $S$.
Take any decomposition $V=V_1\oplus V_2\oplus ...\oplus V_k$ of $V$ into irreducible subrepresentations.
Take any decomposition $W=W_1\oplus W_2\oplus ...\oplus W_l$ of $W$ into irreducible subrepresentations.
For any irreducible representation $P$ of $G$, we have $\left(\chi_P,\chi_V\right) = \left(\text{the number of }i\in\left\lbrace 1,2,...,k\right\rbrace \text{ such that }V_i\cong P\right)$ and $\left(\chi_P,\chi_W\right) = \left(\text{the number of }j\in\left\lbrace 1,2,...,l\right\rbrace \text{ such that }W_j\cong P\right)$. Since $\chi_V = \chi_W$, we thus have
$\left(\text{the number of }i\in\left\lbrace 1,2,...,k\right\rbrace \text{ such that }V_i\cong P\right)$
$= \left(\chi_P,\chi_V\right) = \left(\chi_P,\chi_W\right)$
$= \left(\text{the number of }j\in\left\lbrace 1,2,...,l\right\rbrace \text{ such that }W_j\cong P\right)$
for every irreducible representation $P$ of $G$. In other words, for every irreducible representation $P$ of $G$, the two lists $\left(V_1,V_2,...,V_k\right)$ and $\left(W_1,W_2,...,W_l\right)$ contain the same amount of entries isomorphic to $P$. In other words, the lists $\left(V_1,V_2,...,V_k\right)$ and $\left(W_1,W_2,...,W_l\right)$ contain the same entries up to isomorphism the same number of times (but of course, not necessarily in the same order). As a consequence of this, $V_1\oplus V_2\oplus ...\oplus V_k \cong W_1\oplus W_2\oplus ...\oplus W_l$. In other words, $V\cong W$, qed.
Nowhere did this use that the $V_i$ are somehow unique. (They are not unique as subspaces of $V$. They are unique up to isomorphism, and that follows either from the Krull-Remak-Schmidt theorem or from the fact that
$\left(\text{the number of }i\in\left\lbrace 1,2,...,k\right\rbrace \text{ such that }V_i\cong P\right) = \left(\chi_P,\chi_V\right)$.)