How can a system be unstable if $L(j\omega)$ is never exactly $-1$?
You cannot make conclusions about the stability of a system by only considering its transfer function evaluated on the imaginary axis $s=j\omega$. Replacing $s$ by $j\omega$ in the transfer function only makes sense for a stable system, otherwise you get a function of $\omega$ that does not describe the system, but another (stable) system.

Let me explain this by an example. Assume a causal system with a transfer function

$$H(s)=\frac{1}{s+a}\tag{1}$$

If $a&gt;0$, the system is stable and by substituting $s=j\omega$, we get its frequency response

$$H(j\omega)=\frac{1}{j\omega+a}\tag{2}$$

The corresponding impulse response is

$$h(t)=e^{-at}u(t)\tag{3}$$

where $u(t)$ is the unit step response. We have

$$H(s)=\int_{-\infty}^{\infty}h(t)e^{-st}dt,\qquad \text{Re}\{s\}&gt;-a\tag{4}$$

and

$$H(j\omega)=\int_{-\infty}^{\infty}h(t)e^{-j\omega t}dt\tag{5}$$

Now imagine that we decrease the value of $a$. The system remains stable as long as $a&gt;0$, but as soon as $a$ becomes $0$, $H(j\omega)\rightarrow\infty$ for $\omega=0$, and the system becomes unstable. If we further decrease $a$, the system remains unstable, but the expression $(2)$ does not go to infinity anywhere. However, the point is that the expression $(2)$ is not the frequency response of the system anymore if $a&lt;0$. The reason is that the integral $(5)$ does not exist anymore. So it is pointless to try to deduce stability or instability from the frequency response of a system, because an unstable system has no frequency response.

What you get if you replace $s$ by $j\omega$ in the transfer function of an unstable system is the frequency response of another system with the same expression for the transfer function, but with a different region of convergence (ROC). The ROC is the region in the complex $s$-plane for which the integral $(4)$ converges. For the causal system in our example, the ROC was given by $\text{Re}\{s\}&gt;-a$ (the ROC is always a right half-plane for causal systems). For an unstable system with $a&lt;0$, the ROC does not include the $j\omega$-axis, and, consequently, $H(j\omega)$ does not exist. In that case, the expression $(2)$ is the frequency response of a different system with transfer function $H(s)$ given by $(1)$ but with a different ROC given by $\text{Re}\{s\}&lt;-a$. That system is stable but anti-causal, and its impulse response is given by

$$h(t)=-e^{-at}u(-t)\tag{6}$$

which is clearly different from $(3)$.

Concerning your second question, note that $e^{\pm j\pi}=-1$, so a phase shift of $\pi$ for certain frequencies implies a sign change, which turns the negative feedback into a positive feedback. This can be compared to the positive feedback in an audio system where a microphone picks up the loudspeaker signal, which is fed back into the amplifier, sent to the loudspeaker, etc.