If you don't make that assumption, you also will get "one-dimensional" representations. For example $G=GL(2)$, assume at two place pseudo matrix coefficient for the Steinberg representation and let the remainders be the characteristic function of $GL_2(o)$. This gives you $(-1)^2$ for the the trivial representation.

If you don't make that assumption, you also will get "one-dimensional" representations. For example, consider $G=GL(2)$ and $F$ global function field, assume at two place pseudo matrix coefficient for the Steinberg representation and let the remaining local test functions be the characteristic function of $F_v^\times GL_2(o_v)$. This gives you then $(-1)^2$ for the the trivial representation. The trace formula is still called simple then (see Gelbart-Jacquet), not really more complicated than the one of Flicker-Kazdhan, but you get for each automorphic 1-dim'l representation $\mu \circ \det$ for $\mu^2 =\chi$ one extra contribution.