we biked the whole way by road to maximize the distance traveled each day,
We are planning a road tour of the Alps this summer
my brother and I are very competitive
Get the road bike you love, and work out how to carry a light load for that 1-2 weeks a year when you'll be touring on it. Cyclosportif / sportive events (eg. l' étape du Tour) use this type of bike, and sound like the type of riding you want to do. Even though you will be racing, look for a bike that is geared to comfort, like the Roubaix or the bikes that Chris_W mentioned, or audax- or cyclosportif-specific designs. The UK has heaps of brands that advertise this.