I don't follow the beef with GP, but this is the procedure to get a front Tektro 559 opened up as wide as possible without tools on a bike of that type
1. Set the brake lever adjusting barrel and lockring as far in as possible but with the cable exit slots all lined up with the exit slot in the brake lever
2. Open the QR on the brake caliper
3. Grab a handful of front brake with the left hand
4. With the right hand grab the cable housing where it enters the brake lever adjusting barrel
5. Release the lever with the left hand and as you release pull the housing out with the right
6. Pivot the housing out and release the cable from the brake lever
Now the brake is open as wide as it can be without tools.
For setups of that type I prefer the FL-750 brakelever because it's even easier to do that maneuver, because there is no brake lever barrel adjuster to get in the way. One in line barrel adjuster is plenty and the Tektro 559 has one at the caliper.
Bill Lindsay
El Cerrito, CA