Ironically enough I used the second example mentioned and that bottle broke in transit.

For wine, the white styrofoam packaging that acts as an insert in a cardboard box is my strong suggestion - I've brought back the spoils of wine tours from all over the world in these with a huge amount of success. I also recommend writing WINE on the side of the box - baggage handlers see FRAGILE all day long on a variety of things that may or may not be, but I've found that actually declaring the contents of the package helps handlers make well-informed decisions about how to handle it and most don't want to deal with cleaning up a leaky box

Back on topic, for the plastic bucket, these can be checked and will likely arrive at the destination, but don't expect the lid to hold and don't expect them to be reusable. Every time I've done this, the plastic bin arrives shattered - there's usually at least one crack in the body of the bin, and the rim has lost random shards of plastic making the lid no longer be retained. If you're trying to move a bunch of linens or something, this may not be a big deal, and sacrificing a $7.99 box to move a large quantity may be a reasonable tradeoff. I find that a cardboard box is a better receptacle in most cases, though.

That styrofoam packing is not nearly as safe as several layers of bubble wrap, where you could in theory bounce the bottle on the floor without damage.

Back on topic: I have seldom had issues using tough cardboard boxes to ship household items. You just need to tape them up securely. Plastic storage bins with luggage belts seems a lot riskier.

Agree on plastic storage bins. Cardboard box can give a bit without the box being totally destroyed. Plastic bin gets hit and there's no give without breaking. Bag conveyor belts are not nice to bags. I've had enough hard sided bags come out damaged would never use a plastic bin.

Yes, it will be linens and pillows mostly. And I would do something different if the 2 bins weren't already there - so I just want to use them and not go looking for cardboard or other solution. If the bins weren't already there, I'd probably look for cardboard.

Would it make sense to try to put the bins into an appropriately sized cardboard box (assuming this doesn't make them too big) if you want to keep the bins for future use? In fact, if they were originally purchased in a box and your kid still has the boxes, use them.

I've used plastic bins before (Sterilite I believe); even shipped a computer in one. Still have a few laying in the shed from shipping. Duct tape two/three separate lines a few times around the entire bin, making sure the tape is tight over the cover and you shouldn't have problems.