Oh, good, a dead horse that's not yet quite dead. (-;
No, seriously (sort of):
'Flogging/beating a dead horse' is not only repeating a familiar topic, but doing so uselessly, hopelessly, with no chance that it will accomplish anything. It's a metaphor for wasted effort. So about the cook and the salt, you would only say 'flogging a dead horse' if the point is that the cook will never, ever change, so that it's obviously wrong and stupid of anyone else even to mention the topic. Which is why it doesn't seem right here.
'Rubbing it in' or 'rubbing salt into the wound' also doesn't seem quite right. That's for an acknowledged mistake, like when the cook has already admitted that he forgot the salt, and that he knows the food tastes awful, and he's said he's sorry, but you keep teasing him about it anyway.
'Take the same line' doesn't seem right either. It just means to argue a point from the same angle as someone else within a discussion, to take a similar approach or point of view, to have a similar opinion. It has no suggestion that the line of argument has already been repeated often or is inherently unwelcome. And it's not really used to try to convince an individual person to do something, but rather for conceptual arguments that involve debate in a group, based on a chain (= line, linear sequence) of cause-and-effect reasoning, like, say, at work, in politics, or in an academic setting.
'Hit the same spot' isn't an English expression in this sense at all; in fact, 'hit the spot' is an English idiom that means something completely different (= jmdm. sehr gefallen), so it's really out of place here. 'Strike the same chord' is wrong here for the same reason, because to strike a chord with someone is to please them, to appeal to them.
And 'All roads lead to Rome' is also completely different: It means you don't have to argue about whether it would be better to buy the cook a cookbook or a video of a cooking class, because either would serve the purpose. So it's doubly wrong here.
So I think I have to mostly agree with Plumpaquatsch, even though I only skimmed hastily, and even though I wish he or she would stick with one nick.
As for the English (if we could get back to that) ...
If you're still trying to convince the cook that undersalting is not a good thing, and one person starts talking and another continues, then the second person could be described as
(saying) more in the same vein
expanding on the same argument
pressing their advantage
pressing home the point
If it's the beginning of the conversation and you are obliged to bring up the topic of salt for the nth time because you think the cook can change, and it is not a mistake that the cook has already admitted, then you could say something like
harping on the same subject again
bringing up the same thing yet again
keeping on nagging about sth.
And if you have already been speaking to the cook at length for several months about undersalting the food, and now you bring up overcooking the vegetables as well, then it could be something like
adding insult to injury
piling (it) on
I'm sorry, several of those aren't actually idioms, so they're not as vivid as the German, but maybe that will at least help us focus on the meaning a little better.