Different makes and models of brifters work differently. Whenever you test ride a bike, make sure you understand how the brifters on that particular bike work!
Tipping your handlebars up is dangerous because it puts your brakes out of reach. If you don't have any physical problems, a well-fitting bike should be comfortable in riding position. Once you find a frame that fits you, you can fine-tune the distance to your brake hoods by changing to a stem with a different length, rise or both; by changing to a set of handlebars with different reach; or by moving your brakes higher or lower on the handlebar bend (but not outside of the bend).
Complete bikes usually don't give you much option to raise the handlebars on the steerer tube. Sometimes you might be able to put a slightly taller spacer under the stem and get a few millimeters.
Have fun with your search!
Speed comes from what you put behind you. - Judi Ketteler