Using only one brake worked well in specific circumstances. Old roadsters and many Schwinns positioned the rider so that all weight was over or near the rear axle. Deceleration transfers weight to front wheel unavoidably and every time, so front brake is normally the primary brake. It just matters a whole lot less if sitting right on top of the brake. Second part of just one brake was that bikes were created and most heavily used in places as flat as Chicago or Amsterdam. Braking on steep downhills does not work well with a coaster, but Chicago doesn't have those. And of course coasters were used on bikes expected to go slow.
You will probably need a few extra washers to make your coaster axle work with a normal road frame. Measure axle and rear fork ends. Modern spokes are a bit longer from spoke head to J-bend and many coaster hubs will have relatively thin steel flanges so take up that length with a washer under each spoke head.