I would go ahead and use whatever was finished. If you want to pull cardboard pieces out, then by all means do so. Don’t let perfection be the enemy of good enough.
On a related note, I build my compost piles right in the garden bed and I don’t care if it finishes or not. Just by being in contact with the ground, the pile makes the soil beneath much more fertile. What decomposes decomposes and I either scatter the rest or I use it to start a new pile.
So in the end, I would say go ahead and use what you have.
Some places need to be wild
Willie Smits: Village Based Permaculture Approaches in Indonesia (video)