You are correct that time invariance of laws does not require energy conservation.
It is a common misunderstanding of the content of Noether's theorem. It is not about "laws of physics being independent of time" implying "physics concept of energy is conserved". Those are simplified colloquial statements that are easy to state and remember, but they are inaccurate. There are two problems with such lazy formulations.
First, I can have time-invariant law of physics that every particle experiences both Hooke elastic force $-kx_k$ and friction force $-\gamma m \dot{x}_k$. Such a law of physics would mean the physical concept of energy is not conserved (due to friction force). You can see this hypothetical physics law is time-invariant, yet energy isn't conserved. So we can't formulate the Noether theorem in such a way - it does not work.
Second, Noether's theorem does not say "physics concept of energy is conserved", it says "there is a corresponding conserved quantity based on the action expression and the transformation that leaves $S$ invariant, its interpretation is on you, it may be energy, or not".
Let us formulate the Noether theorem for the special case of time coordinate shift invariance. We need an expression for the action integral, for example the expression
$$
S(t_0) = \int_{t_0}^{t_0 + \Delta t} L~ dt'
$$
where $L$ is some function of coordinates and their derivatives. Noether's theorem states that if $S$ has the "symmetry" that time translations do not changes it (read: $S$ is invariant with respect to change in the time coordinate $t_0$):
$$
\frac{\partial S}{\partial t_0}(t_0) = 0 ~~~\forall t_0,
$$
then the quantity
$$
Q = \sum_n\frac{\partial L}{\partial \dot{q}_k} \dot{q}_k - L
$$
does not change as the system evolves in time - it is a "constant of motion".
However, neither Noether's theorem nor other general rule say this quantity has to be energy (in the physics sense)!
There are examples where the quantity $Q$ is energy, such as the Lagrangian of an harmonic oscillator
$$
\frac{1}{2}m\dot{x}^2 -\frac{1}{2}kx^2
$$
and other examples, where it is not energy, like the Lagrangian of damped harmonic oscillator (Havas' Lagrangian [1]):
$$
L = \frac{2m\dot{x} +kx}{x\sqrt{4mK-k^2}}\tan^{-1}\left(\frac{2m\dot{x} + kx}{x\sqrt{4mK -k^2}} \right) -
$$
$$
-\frac{1}{2}\ln (m\dot{x}^2 + kx\dot{x} + Kx^2)
$$
So, to round this up, Noether's theorem says that existence of symmetry of action implies existence of the corresponding conserved quantity, period, full stop. Interpretation of these quantities is sometimes simple (total energy), sometimes not (Havas' Hamiltonian, which is conserved in time, and thus can't be total energy).
[1] Havas P., The Range of Application of the Lagrange Formalism - I,Nuovo Cim. 5(Suppl.), 363 (1957)