This is a part of an answer; not yet a full answer.

The issue with the question as I've framed it is that this definition of dynamics may not be what a lot of physicists have in mind, even though it is a very intuitive one because it basically says that a dynamical or physical law is something you can use to predict the future from the present, given the present state and how far ahead you want to go.

In my comments under @Andrew's post I discussed this part a bit, and the problem is somewhat subtle: the phase space is the present state of the system, however we may conceptualize a history generated by applying one $\Phi$ family for a certain range of times, then apply another after that, and then another, and so on and so forth. This would not be a dynamical system by the definition given, but would represent a "changing law of physics" in exactly the way we think of via the contrapositive of Noether's theorem, given that we take $\Phi$ to be the object that makes precise the notion of the "law of physics".

Hence, to discuss Noether's theorem in both temporal and spatial translation symmetry, we need to use a bit broader definition of dynamical system. One potential definition to capture the above is that the map family $\Phi$ has both a time increment and a starting time: we might change the notation to $\Phi_{t, \Delta t}$ instead, to emphasize it's no longer a straightforward iterative process. The semantic meaning of this map is "interpret the passed physical state as holding at the specific time instant $t$. Then evolve it forward $\Delta t$ according to the relevant laws of the time." And we require the following: again,

$$\Phi_{t,0}(P) = P$$

but now we modify the semigroup law to

$$\Phi_{t+\Delta t,\Delta s}(\Phi_{t, \Delta t}(P)) = \Phi_{t, \Delta t + \Delta s}(P)$$

where we note now the further evolution by $\Delta s$ has to be started explicitly at the time point where that the first evolution by $\Delta t$ left off, i.e. $t + \Delta t$ from the initial time $t$ at which state $P$ is valid. That part is needed because the laws may have shifted by that point.

In this formalism, temporal translation symmetry can be defined as stating that $\Phi_{t,\Delta t}(P)$ is independent of the starting time $t$, and thus now no longer is a trivial statement - we can have cases where it fails. Or to put it another way, temporal translation symmetry is in effect the statement that the dynamics can be captured with a system of the type described earlier, on a particular phase space.

That said, it still does seem possible we can enlarge the phase space, and then once more we're at the same problem. However - this is where the answer by @Qmechanic comes in: energy is not just any old random quantity we can make conserved with time. It is a very specific such conserved quantity - and that would require further explication and is where my answer must reveal its partiality.

This is a part of an answer; not yet a full answer.

The issue with the question as I've framed it is that this definition of dynamics may not be what a lot of physicists have in mind, even though it is a very intuitive one because it basically says that a dynamical or physical law is something you can use to predict the future from the present, given the present state and how far ahead you want to go.

In my comments under @Andrew's post I discussed this part a bit, and the problem is somewhat subtle: the phase space is the present state of the system, however we may conceptualize a history generated by applying one $\Phi$ family for a certain range of times, then apply another after that, and then another, and so on and so forth. This would not be a dynamical system by the definition given, but would represent a "changing law of physics" in exactly the way we think of via the contrapositive of Noether's theorem, given that we take $\Phi$ to be the object that makes precise the notion of the "law of physics".

Hence, to discuss Noether's theorem in both temporal and spatial translation symmetry, as a non-trivial statement, we need to admit that the definition of dynamics we gave was, in a sense, too strict. One potential looser definition to capture the above is that the map family $\Phi$ has both a time increment and a starting time at which said increment is to be applied: we might change the notation to $\Phi_{t, \Delta t}$ instead, to emphasize it's no longer a straightforward iterative process. The semantic meaning of this map is "interpret the passed physical state as holding at the specific time instant $t$. Then evolve it forward $\Delta t$ according to the relevant laws of the time." And we require the following: again,

$$\Phi_{t,0}(P) = P$$

but now we modify the semigroup law to

$$\Phi_{t+\Delta t,\Delta s}(\Phi_{t, \Delta t}(P)) = \Phi_{t, \Delta t + \Delta s}(P)$$

where we note now the further evolution by $\Delta s$ has to be started explicitly at the time point where that the first evolution by $\Delta t$ left off, i.e. $t + \Delta t$ from the initial time $t$ at which state $P$ is valid. That part is needed because the laws may have shifted by that point.

In this formalism, temporal translation symmetry can be defined as stating that $\Phi_{t,\Delta t}(P)$ is independent of the starting time $t$, and thus now no longer is a trivial statement - we can have cases where it fails. Or to put it another way, temporal translation symmetry is in effect the statement that the dynamics on the given phase space forms a dynamical system as we defined it earlier.

That said, it still does seem possible we can enlarge the phase space, and then once more we're at the same problem. However - this is where the answer by @Qmechanic comes in: energy is not just any old random quantity we can make conserved with time. It is a very specific such conserved quantity - and that would require further explication and is where my answer must reveal its partiality.