I would also challenge your notion. It is true that wealth is correlated with educational achievement. But educational achievement will not, in itself, lead to someone speaking in a way that is more patrician. Is it the kind of society we want to create where you are essentially "locked in" to your parents' social position? It may even be the case that, assuming two people with the same education, the person from a lower class background is more likely to be exceptional.
Just like it is not necessary for the upper class to be more educated, its not necessary for the more educated to speak upper class. But it is more likely.
That such would be enough to stagnate social mobility seems like pessimistic hyperbole to me. Given that we are talking about high value added, high paying, i.e. worker shortage jobs and that there are significantly more demand for such workers than the upper class could provide, meaning, even at worst, such behaviour could only slow down social mobility, not stop it.
If the people who are evaluated are drawn randomly from the population as they claim to in the study, then no. If its a real hiering situation, then possibly yes, though it depends on how the position was advertised. That however may not even change the outcome of the study. In most big companies recruiters are not looking for the best applicant for the job, or atleast have no incentives to do so. They are looking for the easiest to motivate hiring for the job. Similarly, their boss may not have ordered them to find the best worker for the job, they likely asked for a good enough worker who causes minimum issues and who if they cause issues were easy to motivate having hired in the first place. Note that I never claimed that upper class speech meant better at the job, I explicitly wrote academic achievement, i.e the kind of thing which makes for an easily justified hire. Imagine having hired Dr House except without a degree, or a slightly below average doctor with an average grade degree. Now one is plain better, but the other will never get you fired. But thats a whole other problem.
No, that is why interviews should look more like the ones Google performs and less like "please describe yourself, hobbies and life experiences"
With regards to minority tenants, institutionalized racism often leads to the case that non-racist agents will have sound evidence based reasons to discriminate. Not based on the properties of the group discriminated, but based on the behaviour of a third party. A decent example is if banks discriminate against a group, then their availability of credit is worse. This can create self perpetuating circles of discrimination, which can be sustained even if all involved organisations and individuals aren't racist. For a hilarious example of something similar, see: https://i.snipboard.io/kdu77.jpg This is a communication and organisational failure, not racism, but its difficult to distinguish, but critically yes, it means it could be.
Non-discrimination and positive discrimination is best viewed as taxes designed to address this kind of problem, and its working. But the habit people have of challenging this kind of thing as a kneejerk reaction is harmful. What you are doing is essentially claiming that their judgements are corrupt or trying to shame by implied association, and that isn't helpful.