Correctling a sign in a simple circuit with parallel resistors
Your problem is here:

However, when I consider the voltages on the rightmost loop, I see
$$v_1 + v_2 = 0 \Longrightarrow i_1 R_1 + i_2 R_2=0$$

As you go around the loop you're entering one resistor at the &quot;+&quot; terminal and the other resistor at the &quot;-&quot; terminal, so you need to give one of these a negative sign.
Then you'd have
$$v_1 + v_2 = 0 \Longrightarrow i_1 R_1 - i_2 R_2=0$$
which is exactly what you expect.