Yes, you can plug the camera cable while it's not attached on the screen. But you don't need that, it's easy to look through the front camera hole: you should be able to see the camera centered in that hole.
What you are describing looks more like the camera is not working OR the camera cable is not making a connection OR something is damaged from the initial repair.
You tried putting back the original cam that was on the old screen? You know you could use 2 new cams and both could be defective.
1/ Put back the old known working camera and reboot your iPhone (press and hold power button + home button for ~12 seconds, then release home button until iPhone restarts).
2/ If still not working, try:
2-A/ Power cycle the iPhone: disconnect the battery, press and hold Power Button for 20 seconds,
2-B/ Then clean the connector on the board where the cam's cable sits with a cotton swab dipped in 90%+ isopropyl alcohol, then reseat the camera cable (gently but make sure it is well connected), reconnect the battery, and test. Don't forget to adjust time and date otherwise the phone will keep searching for a signal.
3/ If none of the above helps, examine that same camera connector on the board, look for anything damaged/corroded/broken/part removed. If you see nothing wrong, you can VERY VERY CAREFULLY remove the black film that surrounds the screen connectors to reveal the small components, and examine them as well. Let us know if you see anything suspicious, upload a picture if you can.
EDIT: the picture you posted is not close enough. So I am not sure, but is the marked component missing?