Ctrl+L works when a program is running. Clear does not.
I can't see how because ctrl+l is a $SHELL / readline shortcut rather than one defined in the terminal emulator. Once you run a program you're forking STDIN control to another process so you'd have to wait for the $SHELL / readline prompt again just like you would if you typed a command / function / alias into the command prompt.
ctrl+l allows to clear when the prompt is non-empty, maybe parent poster meant that
Sure, theoretically terminal emulators could capture and even rebind those keys via the APIs of whatever graphical toolkit they're built in....but if you wanted to rebind SIGINT to another key in the terminal emulator, that terminal emulator would still have to transmit ^c to the tty.
As for rebinding those keys in the kernel, Linux simply doesn't support doing that and nor does it support binding other keys to different signals. In fact I've tried to do this on a tool I was working on to emulate BSD's SIGINFO in Linux (turned out not to be possible) as as well part of the job control (SIGSTSP et al) support in my $SHELL (https://github.com/lmorg/murex).
This is also why you can throw signals over an SSH (or other remote shell) session when the terminal emulator itself would have no knowledge of the commands running on the remote host.
With respect to rebinding, what is "stty intr ^D" doing if not rebinding the interrupt character to ^D?
I feel your pain about not having SIGINFO though.