As answered in details at https://superuser.com/questions/1164104/semi-transparent-colors-in-terminal, palette colors cannot contain alpha channel. Only the default background can be transparent.
(Just for the record, recent versions of mainstream gnome-terminal don't have transparency support at all. Older versions used to have it, and some distributions, including Ubuntu, patch it to bring it back.)
"OpenSSH dutifully sends ANSI background colors" – I seriously doubt it. ssh's responsibility is to send a stream of data through the wire in a way that it arrives at the destination exactly as it was sent by the sender (after encryption on the sender side and decryption on the receiver side, of course). It is not a terminal emulator, has no idea whatsoever about escape sequences, ANSI color codes and such. It's not even guaranteed that whatever ssh sends will make it to a terminal emulator on the receiving end. E.g. think about scp, it would be weird if you couldn't safely copy files that contained some ANSI escapes. ssh surely leaves your data unchanged.
It must be some shell setup or similar on your remote host that decides to use colors, and if you had the same setup locally it would be just as broken for you without ssh in the game. You should locate who and where emits this sequence as part of your login procedure or your prompt.
Although not clear from your screenshot and your description, I suspect that non-transparent gray lines start to appear when you first reach the bottom of the terminal and the contents start to scroll. This is due to the braindamaged "bce" (background color erase) misfeature of most graphical terminal emulators. gnome-terminal (in fact the underlying vte widget) has changed its behavior as of vte-0.44, which appeared in Ubuntu Yakkety. In case you're running Xenial, you might want to give it a try: update your lilbvte-2.91-0 package from Yakkety, quit all gnome-terminal windows and start again. Presumably you'll see a different experience, one that you wish to see. Note, however, that gnome-terminal's (vte's) new behavior is not standard conforming, and you'll still see the old behavior in other emulators, e.g. xterm. Details about the gnome-terminal change around the bce feature: https://bugzilla.gnome.org/show_bug.cgi?id=754596.