How do I connect to a websocket manually, with netcat/socat/telnet?

I think you want to modify the socket stream to translate \n (line feed) to CRLF (Carriage return & line feed). Doing info socat produces detailed information which includes this modifier:

crnl   Converts the default line termination character NL ('\n',  0x0a)
       to/from CRNL ("\r\n", 0x0d0a) when writing/reading on this chan-
       nel (example).  Note: socat simply strips all CR characters.

So I think you should be able to do this:

socat - TCP:echo.websocket.org:80,crnl