You're misrepresenting the actual position, which is: If we detect EHG ancestry at the proposed language shift you have evidence for the Steppe Hypothesis yet if you don't you don't have evidence against it. It's a form of this rule: Absence of proof isn't proof of absence.
The beauty of Ancient DNA is the ability to tell us where ancient people came from, and how they relate to the people living today.
When asking the question of where did the Anatolian speakers like Luwians and Hittites come from, you have test samples before and after they're arrival, assuming that you know when they arrived, that's no problem if you have a logical sequence of samples throughout the archaeological record.
We know that the people who carried IE languages to Europe arrived from the Steppe because that was the difference compared to the previous era.
Where did the Steppe people come from? a mixture of two populations, EHG and a southern population from Caucasus, logically, one of them is ultimately the origin of the deepest language that's the origin of all indo European languages.
The brute force method is to test every IE speaking population in the world, but you can also use linguistics as a guide, the Anatolian branch is the most diverging and shares the least with the others, and so their DNA will solve the question.
Now we have Ancient samples, and the only difference is that they have CHG, but not EHG, so some said they are not Anatolians because they don't have EHG ? This belief of EHG = IE has not yet been established for it to be used as an argument.
We were in process of testing it, a test can fail, otherwise, you don't have a hypothesis, you have a belief.
The hypothesis that only the elites can have EHG, is still within the domain of logical possibility, but probabilistically unlikely, the cruelest and gene flow prohibitive caste system that I can think of is that of Hinduism, and even there the lowest castes have Steppe ancestry, and the highest castes have ASI ancestry, it is still detectable.