You are coming to a wrong conclusion due to a misunderstanding of what the separation of powers means. I will first try to illustrate with a thought experiment, after which I believe you will agree that there is something wrong in your reasoning, and then I will demonstrate where your logic went wrong.
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Thought experiment: Suppose a dear loved one is brutally murdered by a relative of the current democratically elected leader (imagine a hypothetical leader, country, etc). Through various extralegal manipulations, the leader ensures that the murderer is not convicted (evidence disappears, jurors are appointed in a fishy way, the judge turns out to be a family member of the murderer, ...) and you notice that none of the usual paths of recourse work. Perhaps you go to the press, but his supporters just dismiss this as a smear campaign. Crucially, this leader is very popular, their party controls the legislative and has appointed judges for years.
Following the reasoning in your post, which I think can be summarized as "the legislative and judicial branches, which are legitimately elected/appointed, chose not to stop him, therefore the separation of powers is not violated and this is how a democracy is supposed to work", the leader's actions do not constitute a violation of the separation of powers, and this incident does not demonstrate that this country's democracy is unhealthy.
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I hope you agree that this conclusion is wrong, yet it follows inexorably from the argument you have made (because the sole precondition, that the other branches are legitimately elected/appointed, is satisfied). So we must conclude that there is a mistake in your argument, and I think it originates in the conflation of two of the core features of liberal democracy -- that is (a) leaders are elected and (b) there is a separation of powers. You are essentially saying that (b) holds because (a) holds, but it is important to remember that (a) and (b)are independent features that sometimes oppose each other: it is by design that the system (especially the judiciary) can overrule the majority of the population, at least for some time.
So the question of "are the judiciary and legislative branches effectively enforcing the separation of powers" is not actually related to whether these branches are legitimately appointed/elected, but to whether they are independent. By this I mean that they play their constitutionally prescribed role even if at times this is unpopular. For example, the judiciary's job is to enforce the law. In the thought experiment, they are not independent from the executive, and that is a deep system failure: they should enforce the law (convict the murderer) even if the (popular and legitimate) executive disagrees.
For example, the law is crystal clear wrt who has the authority to enact tariffs on foreign nations. The President cannot legally do this as the Constitution vests the power to raise taxes in Congress; reasonable people cannot disagree about this. Congress has granted him emergency powers on the basis of a fentanyl crisis at the Mexican border; the scope of these emergency powers clearly does not include imposing tariffs on, say, Australia. Again, there is no room for interpretation here, this is all crystal clear. The fact that the tariffs haven't been effectively struck down yet is a clear failure of the separation of powers, because the law is so clear. The popularity of the president or his policy is completely irrelevant to the question of whether he should be stopped by the courts.
The main reason this needs to exist is to make sure that, indeed, the next election is a free and fair election. If the separation of powers does not hold, then there is nothing stopping the executive from manipulating the election and hollowing out democracy. This has happened many times in history, and it is exactly what people (rightfully, I believe) fear about the Trump presidency.