Here's a ground-hugging Flowering Dogwood with all the beauty but none of the issues!  
 
Bunchberry Dogwood thrives in moist, well-drained soil in dappled sunlight to full shade.   Its flowers, four white bracts opening up to reveal a small green flower in the center, look just like Flowering Dogwood's, except they're of a size suitable to the ground-hugger.
 
Birds love its bright-red drupes, which ripen in August, and fall foliage is a deep purple-red.
 
Bunchberry spreads by underground stems and makes an excellent groundcover for a shady area.  Bunchberry can handle full sun in cool summer climates.