I am a NY lawyer. Without seeing the letters, in general there is no reason you can't open an estate bank account with regular letters - the size is not relevant - with a national bank - like Chase or Citibank (not that I am endorsing any bank in particular) that has a branch also in NY. (Look at your oath and designation) If the letters are restricted, this is a different story and would depend on the exact restriction. In person is not relevant - just ask for the forms, give the tax id, letter of instruction, and once you mail the required info., there should be no need for you to go in person. I have clients that I have never met, but have opened estate accounts for them - even international.
In order to distribute directly from the brokerage account, you likely would first have to transfer the proceeds (even at the same brokerage) to an estate account. Then depending on the terms of the will, you could distribute from the estate brokerage account.
No, you can't deposit the money into your account - this is comingling of funds and is a breach of your fiduciary duty.
You can't appoint somenone to be your agent as executor. If you can't act, perhaps you should resign. Good luck.
Sharon M. Siegel, Esq.
Siegel & Siegel, P.C.