Coax through wall box
Yep, especially if you can see the end is free. If you think there's enough in the box to get a connector on, then just cut it and activate the box.

If you'd like to try and get more wire, then use pliers with their teeth wrapped a couple of times with electrical or duct tape. As long as you don't try too hard and rip the wire open, you won't damage anything.

Likely, you just need to break the foam's initial sticky area to slide the wire. Go straight in at the top or bottom of the box (top if pulling down and bottom if you're pulling up).

Use the outer edge of the box as your lever's (pliers) folcrum and pull up or push down (see above) the plier handles while gripping the wire tightly. This will usually provide much greater power and pop the foam's seal.