Teager-Kaiser Operator vs. Hilbert Transform
This paper may be of interest:

David Vakman, "On the Analytic Signal, the Teager-Kaiser Energy Algorithm, and Other Methods for Defining Amplitude and Frequency." IEEE Trans. Signal Processing. (1996)

Summarising from the paper:

$$\Psi(u) = a^2w^2 = [u'(t)]^2 - u(t)u''(t)$$
$$\Psi'(u') = a^2w^4 = [u''(t)]^2 - u'(t)u'''(t)$$

where $u(t)$ is the signal, and $a$ and $w$ iare the amplitude and frequency estimates to solve for.

$$a(t) = \frac{\Psi(u)}{\sqrt{\Psi(u')}}$$
$$w(t) = \sqrt{\frac{\Psi(u')}{\Psi(u)}}$$

But this means if at some point in the signal $u'(t)=1$ and $u''(t)=u''''(t)=0$, for example, then $\Psi(u) = 1$ and $\Psi'(u) = 0$ and thus $a(t) = \infty$ and $w(t) = 0$.

The paper thus explains this causes spikes in amplitude and zero frequencies with the Teager-Kaiser algorithm.

For your signal, the reverse must be true. If $\Psi(u) = 0$ and $\Psi'(u) = 1$ then you would get spikes in frequency when amplitude is zero. This would occur when $[u'(t)]^2 = u(t)u''(t)$

The Hilbert transform uses the entire signal to compute. It is implemented by doubling the positive frequencies and setting the negative frequencies to zero. Therefore the energy of the signal does not change (Parseval's theorem). Thus, intuitively, there can be no infinite spikes, as this would require infinite energy.