While I understand caution, and urge the OP to do whatever creates the greatest feeling of comfort, I still have seen nobody provide any information that would make me understand that a cardboard box provides any more protection than a moving blanket.

If cardboard is that much better, then we should insist that our antique tables are put in a rough abrasive box rather than a nice padded moving blanket. Remember, these companies use moving blankets, not bedding... They are more padded, heavier and more durable than your bedding. What forces could be applied to a biycle that would damage it while being properly wrapped in a moving blanket that would be prevented by a cardboard box?

With all that said, I do like the idea of taking the wheels off the bike, as t does seem to minimize risk in my mind.

Also, unless you know for sure that the same truck will be used for the whole trip (unlikely) don't trust that something will be kept upright and not laid down when repacked in another truck... Pack it to be safe in any orientation.

Take pictures of items that are unique, and don't ship collections or other small valuable items... any reputable moving company will warn you about this.

Finally, I had nothing missing from my trip, and the one case where I heard someone had something lost, he got reimbursed when he persisted, a pain, but it worked. When packing, make sure that the description of the items is accurate, and as someone else noted, check off everything that is unloaded based on that list.