I agree that it's a sort of UI problem. I could be sharing a desktop with a customer with Teamviewer (which maybe has file sharing, and it's telling I'm unsure about it) and at the same time talking with him on Skype, which for voice is still good (it's bad for everything else - text, files, screen sharing - video calls are of very limited use.) But I can't easily send a file to the other guy.

Unfortunately AFAIK there has been no successful open point to point file transfer protocol that different OSes could implement and be interoperable over the Internet. Going to https://send.firefox.com/ and drop a file there is not an improvement. It's still centralized. Is there any solution to the problem of discovering the address of the sender and the recipient without a central server? I would think it's an impossible problem but there are clever people out there. Maybe mesh networks?