Flat bars are not going to be causing problems on 10-mile rides, unless there is a problem with the way the bike is set up or it is grossly mis-sized for the rider. I'm routinely doing 20-25 miles a day including a fair bit of bumpy gravel on a rigid frame with flat bars and the most I ever get is a mild discomfort that just requires a quick shake of the hand to alleviate.
Bike set-up certainly needs to be considered, but the overall fit is important. For example, a LOT of cyclists have their saddle set too low, as they have this weird thing about putting their feet flat on the ground whilst seated - which makes riding hard work and is very fatiguing on the legs. Therefore having the bars relatively higher than the saddle may not be a good outcome if it means the saddle gets lowered below it's optimum.
When I get on a new-to-me bike, the first thing I do is set the saddle height, and I generally also try to get the fore-aft adjustment so the seatpost comes up through the centre of the saddle as a starting point. This can't be over-emphasised; the saddle height is THE single most important setting that you can make on a bike, and everything else you alter should always be done AFTER you have got the saddle correct. Once I know my legs will propel the bike effectively as possible, I turn my attention to the handlebar height, and whether the reach to the bars is comfortable or too stretched.
Obviously, if you are riding an off-the-peg factory-produced bike, all it's dimensions will have compromises to give an acceptable fit to the "average" rider. That means sometimes you can jump on a bike and if really lucky, it will fit like it was made-to-measure, or more likely it will sort-of fit but need some fiddling around with to optimise the comfort. Often you will read comments about getting bike fits; I am very much of the opinion that 99% of bike set up issues can easily be resolved by the rider themselves having a methodical approach to how they adjust their bike to suit their own body proportions.