Bear boxes are containers to store/protect food that can be smelled/eaten by wild animals. Bears, wolves, coyotes, etc. They help to minimize contacts between people and animals at inappropriate times. I think hunters use bear boxes. If a hunter kills a deer, or brings a day or two of fresh food with them when camping, then they will put the raw/fresh food/kill in the bear box at night. Very few bikers carry raw food with them. Do you cycle with raw meat? Most bikers who are cooking their food will carry rice or noodles or cheese or cans of tuna or oatmeal. Many just buy the food they are going to cook that night and carry no food while riding and have no food left overnight. I don't think Gu packs or energy bars or granola bars or fresh apples or bananas are foods that bears or coyotes are attracted to. Bikers have these foods 24 hours a day usually. On your biking travels, what foods do you have overnight in camp that would need to go in a bear box?
And bears are not very plentiful in the US. They are very rare. There are only a few places in the US with any bears.
On the notion that bears are mostly interested in raw meat... That is just plain wrong. Black bears are omnivores and additionally in some places have become habituated to stealing human food and they like pretty much any of it including cans. In the Sequoia, Kings Canyon, and Yosemite parks they break into cars where the see pretty much any human food. It is illegal to not put all food and scented products like toiletries in the bear boxes. The ranger tells you when you enter the park, they make rounds reminding folks, and they write tickets with fines attached for violators.
For the entire western states, that is all of the states with the Rocky Mountains to the Pacific Ocean, it averages out to one black bear per 7.8 square miles. About 2.8 miles by 2.8 miles. One bear in this area. 150 thousand bears in about 1.2 million square miles. In California, Idaho, Oregon, and Washington, there are fewer square miles per bear.
What constitutes plentiful is debatable, but those numbers and my experiences don't line up with your "very rare" and "only a few places in the US with any bears" comments. Yes I would stop short of saying they are plentiful, but I'd also wouldn't call them very rare either.
Depending on where you tour goes the chances of seeing a bear may be anything from non-existent to fairly likely. For example I think that if you were to ride the Sierra Cascades route or even just the Yosemite, Kings Canyon, Sequoia portion you are probably more likely to see a bear than not. In that section I think we saw 6 different bears in 4 different locations. I saw bears there when I was backpacking as well.
Doing the TA, it isn't all that unusual to see a bear, but yes it is probably more likely that you won't.
So bottom line most folks won't see a bear when on tour unless in places where they are most common,but failing to use bear boxes where they are supplied would be foolish, illegal in many places, and people who don't use them are causing bears to become human habituated and ultimately be put down.