i can understand the feeling of starting with a tree that you cut down and ending up with a finished bow from it but...
wood is wood and a bow is a bow. a board bow doesn't have to be "dimensional" nobody has ever said you can't chase a ring and follow the grain on a board bow, doing this can result in a snakey character bow just as easily as a stave can.
Cheaper to cut staves? maybe maybe not, if the tree is in your backyard then yes, do you have to drive to some wood lot, gas isn't free. chainsaw? i don't have one. on the other hand there is a hardwoods dealer 2 blocks from my place of employment, in my case board bows are way cheaper than staves, unless my generous friends give them to me
now having said all that, i prefer staves, its just not always the best option for me when i have to consider cost, time, etc.