Chase was the one I had to go to the US Consulate for. I've also used TD Bank, and some others that I don't remember.
The key thing is that a lot of their front line staff may either not know or don't want to deal with the hassle of this for a "normal" account holder.
Best bet is generally to get through to whatever department handles high net worth individuals or corporate accounts. Some banks also have special departments handling expats that are used to dealing with the hassle of people who are not present. And even if you're a foreign national, those departments may be able/willing to help.
Even if they're not the right ones to handle the account opening (I certainly was not a high net worth client when I opened my first ones) these are the teams most likely to know if their bank allows for it and who to talk to.
If you have a lawyer or accountant, my impression is that your odds go up just by channeling your request through them. Presumably it makes them think you're more likely to be a potential high value client. In general this sidesteps a lot of complications with banks.
An approach some may allow if they do need you to come in is for you to come in to non-branch offices. E.g. a large bank with a private wealth management or investment banking office in your country may be willing to let you pop in (by appointment) to show your ID there if you find the right person and ask nicely.
I get the impression that with a lot of banks just demonstrating that you know about premier/high net worth departments or think to go to a corporate / investment department creates the implication in their mind that you're not a regular customer and should be treated with some extra care just in case you turn out to be important.