When did the phrase "objective standpoint/objectively" start getting used in criticism even though inherently, there is no such thing as objective criticism?

There are two questions here. One is about the phrase 'objective criticism', and the second is the one captured in your question:

When did critique get reduced to like this?

This second question is easier to answer, because the first question is fraught with different interpretative questions. These interpretative questions arise as a result of the ambiguity of 'objective'. The word has a lot of different meanings. You obviously don't have in mind many such meanings. For instance, 'objective' can mean the same as 'mind-independent' or 'human-independent'. Like, when people are debating whether beauty is socially constructed or objective in the sense of being independent of human beings. However, 'objective' as you mean likely means something like value-free, being faithful to facts, the absence of normative commitments, free from personal biases, or free of any particular perspective. These are all distinct, competing definitions of the word 'objective', and it is very important for the purpose of answering this question that we not treat them all as the same. But this very fact about the word 'objective' makes the first question much harder to answer. I will circle back to this question in a bit.

The second question, about when objective criticism came into existence, is easy to answer because it's existed (at least in the West) for quite a long time, our record shows. Plato -- who is the only Greek philosopher from antiquity whose works we have in full, besides the later Platonist, almost a millennium later, in fact, Plotinus, -- provides objective-style criticism of Homer's works. Roughly 1.5 books of his masterpiece the Republic is devoted to criticism of Homer's portrayal of the gods (and the portrayals by similar poets). Earlier thinkers did the same, but it is difficult for us to say a lot about them because their works don't come down to us in full, but here are two relevant fragments from Xenophanes:

Homer and Hesiod have attributed to the gods all sorts of things that are matters of reproach and censure among men: theft, adultery, and mutual deception. (B11)

…as they sang of numerous illicit divine deeds: theft, adultery, and mutual deceit. (B12)

What we can learn from this is that philosophers and other thinkers strongly disagreed with your view that "there is no such thing as objective criticism."

After all, why would think such a thing is true?

It might occur to you that nobody is ever "objective" because we are all "biased" or have a "perspective." And virtually every philosopher would push back on you for saying this for a bunch of reasons. Some would say you are wrong on the grounds that we aren't always biased. Others because it is possible to eliminate biases. Others because being biased doesn't mean you can't be objective.

There are so many things we can about why historically, people disagree with your view. But here are two ways to go about this.

The first is to make a distinction between discovery and justification. This distinction is owed to Reichenbach. He first made this distinction with respect to the epistemology of mathematics:

the objective relation from the given entities to the solution, and the subjective way of finding it, are clearly separated for problems of a deductive character […] we must learn to make the same distinction for the problem of the inductive relation from facts to theories. (Reichenbach 1938: 36–37)

The standard interpretation of this statement marks contextual values, which may have contributed to the discovery of a theory, as irrelevant for justifying the acceptance of a theory, and for assessing how evidence bears on theory—the relation that is crucial for the objectivity of science. Contextual values are restricted to a matter of individual psychology that may influence the discovery, development and proliferation of a scientific theory, but not its epistemic status.

For instance, maybe I discovered a theory because I was motivated by some strong emotion or because I was terribly biased against someone or some view or maybe I just got lucky and guessed it based on some tea leaves. This would mean that my discovery was not at all "objective." But this doesn't say anything about the justification of my theory -- namely, how well-supported it is. So, in this sense, everything that mattered was perfectly objective. It doesn't matter at all that I was a deeply biased, flawed researcher. All of that stuff is true but perfectly irrelevant to how justified my theory is. The same goes for "objective criticisms": maybe I am an idiot or someone dominated by feelings, but my criticism stands on its own.

The other approach is the one favored by philosophers historically like Plato and Xenophanes. We see it best of all illustrated in theories of measurement.

Measurement is often thought to scientific objectivity. The most famous statement is Lord Kelvin's dictum:

when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be (Kelvin 1883).

Measurement can certainly achieve some independence of perspective. Yesterday's weather in Montreal, Canada may have been “really hot” to the average Canadian and “very cold” to the average Mexican, but they'll both accept that it was 21°C. Clearly, however, measurement does not result in a “view from nowhere”, nor are typical measurement results free from presuppositions. Measurement instruments interact with the environment, and so results will always be a product of both the properties of the environment we aim to measure as well as the properties of the instrument. Instruments, accordingly, do provide a "perspectival", "biased" view on the world (cf. Giere 2006).

Moreover, making sense of measurement results requires interpretation. Consider temperature measurement. Thermometers function by relating an unobservable quantity, temperature, to an observable quantity, expansion (or length) of a fluid or gas in a glass tube; that is, thermometers measure temperature by assuming that length is a function of temperature: length = F (temperature). The function F is not known immediately, and it cannot be tested either (because it could in principle only be tested using a thermometer, and the accuracy of the thermometer is just what is at stake here). Making a specific assumption, for instance that F is linear, solves that problem by just saying so. But this “solution” does not take us very far because different thermometric substances (e.g., mercury, air or water) yield different results for the points intermediate between the two fixed points 0°C and 100°C, and so they can't all expand linearly.

In Hasok Chang's account of early thermometry (Chang 2004), the problem was eventually solved by using a “principle of minimalist overdetermination”, whose goal it was to find a reliable thermometer while making as few substantial assumptions (e.g., about the form forF) as possible. It was argued, eventually, that if a thermometer was to be reliable, different instances of the same thermometer type should agree with each other, and the results of air thermometers agreed the most. “Minimal” doesn't mean zero, however, and indeed this procedure makes an important presupposition (in this case a metaphysical assumption about the one-valuedness of a physical quantity). Moreover, the procedure yielded at best a reliable instrument, not necessarily one that was best at tracking the uniquely real temperature (if there is such a thing).

You might wonder how this translates to objective criticism and to, as I said, what Plato and Xenophanes said. The answer is that Plato and Xenophanes leveraged this criticism by doing a non-mathematical version of this process. They made it clear what exactly they were aiming to show and then showed it. So, instead of just saying 'this poem is bad, because it depicts the gods incorrectly', they said: 'when I say that this poem is bad, I mean that it gets at this important thing incorrectly, and here's why it this matters, and here's how the poem does this.' In other words, they avoided the charge their criticisms were "subjective" because they outlined exactly what their terms meant, could defend the meanings of their terms in case of disagreement, and then showed that their criticisms followed, when you accept the meanings of their terms in addition to the evidence they were gathering. This is effectively how measurements we work: we establish a baseline, and then use evidence to show how things stand with respect to the baseline. The only reasons why we ever doubt that a criticism is objective are 1) either the person who is presenting the criticism is doing a bad job explaining their evidence or the meanings of their terms, or 2) we are just not paying careful enough attention to the fact that when they say something is 'bad', they actually have a carefully-defined account of what badness in, say, artwork is. Often people give up before they even listen to the criticism and think that badness, or beauty, or something, is just by its very nature "subjective" -- but it's not even clear what on Earth that means, let alone what evidence there is that it is true.