Is there an actual proof for the energy-time Uncertainty Principle?
The main problem is, as you say, that time is no operator in quantum mechanics. Hence there is no expectation value and no variance, which implies that you need to state what $\Delta t$ is supposed to mean, before you can write something like $\Delta E \Delta t\geq \hbar$ or similar.
Once you define what you mean by $\Delta t$, relations that look similar to uncertainty relations can be derived with all mathematical rigour you want. The definition of $\Delta t$ must of course come from physics.
Mostly of course, people see $\Delta t$ not as an uncertainty but as some sort of duration (see for instance the famous natural line widths, for which I'm sure there exist rigorous derivations). For example, you can ask the following questions:
Given a signal of temporal length $t$ (it takes $t$ from "no signal" to "signal has completely arrived"), what is the variance of energy/momentum? This can be mapped to the usual uncertainty principle, because the temporal length is just a spread in position space. It is also related to the so-called Hardy uncertainty principle, which is just the Fourier uncertainty principle in disguise and completely rigorous.
If you do an energy measurement, can you relate the duration of the measurement and the energy uncertainty of the measurement? This is highly problematic (see e.g. the review here: The time-energy uncertainty relation. Choosing a model of measurement, you can probably derive rigorous bounds, but I don't think a rigorous bound will actually be helpful, because no measurement model probably captures all of what is possible in experiments.
You can ask the same question about preparation time and energy uncertainty (see the review).
You can ask: given a state $|\psi\rangle$, how long does it take for a state to evolve into an orthogonal state? It turns out that there is an uncertainty relation between energy (given from the Hamiltonian of the time evolution) and the duration - this is the Mandelstamm-Tamm relation referred to in the other question. This relation can be made rigorous (this paper here might give such a rigorous derivation, but I cannot access it).
other ideas (also see the review)...
In other words: You first need to tell me what $\Delta t$ is to mean. Then you have to tell me what $\Delta E$ is supposed to mean (one could argue that this is clear in quantum mechanics). Only then can you meaningfully ask the question of a derivation of an energy-time uncertainty relation. The generalised uncertainty principle does just that, it tells you that the $\Delta$ quantities are variances of operators so you have a well-defined question. The books you are reading seem to only offer physical heuristics of what $\Delta t$ and $\Delta E$ mean in special circumstances - hence a mathematically rigorous derivation is impossible. That's not in itself a problem, though, because heuristics can be very powerful.
I'm all in favour of asking for rigorous proofs where the underlying question can be posed in a rigorous manner, but I doubt that's the case here for a universally valid uncertainty relation, because I doubt that a universally valid definition of $\Delta t$ can be given.
This is the case. The uncertainty relationship with energy and time is a matter of Fourier analysis. In fact the relationship $\Delta\omega\Delta t \simeq 1$ was know in classical EM and electrical engineering before quantum physics. The use of Fourier analysis in electrical engineering had much the same uncertainty relationship as the reciprocal relationship between frequency and time.
Classical mechanics has Poisson bracket relationships between momentum and position, and quantum mechanics has an operator replacement of these $$ \{q, p\} = 1~\rightarrow~[q, p] = i\hbar. $$ There are no Poisson brackets in Hamiltonian mechanics between time and energy. In quantum mechanics there is by corollary, using the word informally, no time operator. This leads to some interesting complexities with relativistic quantum mechanics and quantum field theory.
Quantum mechanics is a wave mechanics, and the Fourier analytical basis for the time-energy uncertainty is "good enough" to accept it. The physical basis for energy=time uncertainty is strong enough to accept. We just have some distinguishable situation between space and momentum vs time and energy. In some sense this is a mark that is contrary to the Einsteinian idea.