You seem to believe that you need to open the connection in WinSCP to open PuTTY from WinSCP. You do not have to. You can open the session in PuTTY straight from the WinSCP Login dialog.
Use Manage > Open in PuTTY:
https://winscp.net/eng/docs/ui_login#opening_in_putty
Or even easier, use Login > Open in PuTTY or Ctrl+P.

I don't use PuTTY, it's far inferior to Windows Werminal or any other SSH client alternative.

So now what I do, is I save some of my sessions in Windows Terminal settings.json, and some of my sessions in WinSCP. I'd like to be able to keep everything in WinSCP and only start the Windows Terminal from WinSCP, if I need SSH access.

But I can't do that, since WinSCP will fail to connect, and can't share sessions with the Windows terminal.

I do not understand. If you use the WinSCP Login dialog as a session manager for PuTTY/Windows Terminal, why would you need WinSCP to support SSH? Just store SFTP site in WinSCP and open it in PuTTY as SSH.

See also Using WinSCP as PuTTY session manager with Folders and Incremental search

On some servers, we don't need SFTP, so it was disabled. We get this error if we try to log in, we get this error,

It would be nice if there was an option to allow only SSH connection and not try to establish SFTP connection so that I don't have to save the session inside PuTTY or Windows Terminal. I would be able to have all my sessions in WinSCP only.