Contacting potential advisor for PhD
What might be very impressive is if you would be willing to apply as a special student.

This might be a good “backdoor” way to get into a top 10 university.

I worked in the office of graduate admissions at Harvard university:

Most do not realize that applying as a special student is just as competitive as applying as a degree seeking student, but in many cases you do not have to take the GRE

Here is the answer I gave to another person on this forum:

https://academia.stackexchange.com/a/61120/123306