You're right that it's not exactly complete. Time-translation symmetry is what would make an experiment "not depend on when the experiment took place," but time-reversal symmetry, another "time symmetry," would mean that it did not matter whether a VCR tape of the experiment was playing forwards or in reverse, so that both directions appeared to be completely reasonable events. (For example a car driving forwards is just as reasonable as a car driving in reverse.) Time translation symmetry appears in real life, but time reversal in general does not.

On the contrary, it is a pretty fundamental assumption in all of our fundamental theories (with some bows and whistles).

A point of confusion might be that the surroundings of the experiment might influence it and the surroundings would depend on whether it is a workday or a weekend, etc. But this is a bad experiment. The correct interpretation is "if my experiment is isolated so that external events do not influence it (like all good experiments), then I will get the same results no matter whether I do it today or wait until tomorrow". Another way to phrase it is "there is no dependence on the variable 't' in the fundamental laws".

This is correct and leads to conservation of energy — to see why, imagine a frictionless pendulum. If you prove that it returns to original position once, you know that it will swing for eternity because if the system is in the same state and only the time has changed, it do exactly what it did previously. Which is to say that the energy of the system will remain constant.