Is it the clip on the main board or daughter board end? They are required to hold the cables in place and push down on them to ensure connection, so yeah if there’s no clip it could stop everything on the left side from working.
I actually have the same problem with my Vita, except it’s the clip on the main board that’s broken and that would obviously be more expensive to replace. I found that if I pushed down firmly on the cable socket everything would work, so I took a small piece of really firm foam and stuck it onto the back of the case with double sided tape, so when the console is assembled that foam pushes down on the cable in the socket and makes the connection.
So I don’t know if that will work in your case, but you could try pressing down firmly on the clipless socket to see if the controls start working, and if so then you could do that or something like it. With larger cables/sockets it’s possible to hold them in place using things like a thin piece of plastic or even paper or cardboard, but the Vita is a bit small for that. Repairing the clip or replacing the board would be the ideal fix though.