The WP summary of the Mandeshtam-Tamm relation is, for an observable $\hat B$,
$$
\sigma_E ~~~\frac{\sigma_B}{\left| \frac{\mathrm{d}\langle \hat B \rangle}{\mathrm{d}t}\right |} \ge \frac{\hbar}{2} ~~,
$$
where the second factor on the l.h.s., with dimensions of time, is a lifetime of the state ψ with respect to the hermitean observable $\hat B$. Roughly, the time interval (Δt) after which the expectation value ⟨$\hat B$⟩ changes appreciably.
For a stationary state, the drift rate of ⟨$\hat B$⟩ goes to zero, and the variance of energy goes to 0 as well, as it should.
This is all in standard QM, unitarily evolving, with or without measurements. You may do any and all measurements discontinuous, delirious, expialidocious, whatever, and plot your results, but you must be talking about the same state ψ all the time. The distribution in B will have a variance, which is what is under discussion.
(Heuristically, a state ψ that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must be defined accurately, and this requires the state to persist for many cycles, the reciprocal of the required accuracy. In spectroscopy, excited states have a finite lifetime. By above, they do not have a definite energy, and, each time they decay, the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fast-decaying states have a broad linewidth, while slow-decaying states have a narrow linewidth.)
I have not fully appreciated your misgivings, but they seem to me to also apply to the standard Δx Δp uncertainty principle: A pure state will have corresponding distributions for x and p with nontrivial variances, computable through standard continuous QM, which your measurements will probe.
