Ok we might as well dive into this deeper...
The reason quality suffers at higher frame rates has more to do with data rate than shutter speed. The GoPro can only record so much information to the SD card per second. To keep the math simple, let's say the max data rate is 60megabits per second. Therefore at 60 fps, each frame can store 1megabit. But if you shoot at 30fps, each frame can be 2megagabits. Double the data, double the quality.
Unfortunately we can't directly control shutter speed on the GoPro. It is set automatically based on the amount of light detected. If it's a really bright day, the GoPro will automatically crank up shutter speed regardless of frame rate. So for example, you could be shooting 30 fps, but during each frame the shutter might only be open for 1/120th of a second. If you changed to 60fps but the lighting stays the same, the shutter would still only be open for 1/120th of a second during each individual frame. So in this example, the shutter speed is the same at both 30fps and 60fps. The image quality will still be higher in the 30fps version because of the data rate.
You can trick your GoPro into using a slower shutter speed by putting on an ND filter. The GoPro thinks it's darker outside than it is, so it keeps the shutter open longer. But if your shooting at 60fps, the maximum the shutter can be open during each frame is 1/60th of a second. If you are shooting at 30 fps, the shutter could be held open 1/30th of second. That allows the sensor more time to soak up light. It can result in a cinematic looking motion blur - without loosing detail or quality in stationary subjects. The motion blur also reduces or removes jello.
If you shoot with Protune on, the GoPro increases its data rate, improving quality... but you better have a fast SD card to keep up. Also, you have to color correct your video in post because Protune footage looks flat and lifeless until it's processed - when Protune is OFF, the GoPro processes the picture for you. With Protune ON, the GoPro captures more data, but leaves the processing up to a human using software on a computer.