A spinning wheel will turn in the direction it leans, which is why when you lean into a corner the bike will steer itself around the corner. Also if you roll a coin on a flat surface it will roll in a circle because as the coin falls it turns in that direction.

If your frame and/or fork is twisted, one wheel is always leaning and the bike will steer in that direction. Even if you do the string test, which is primarily to align the rear triangle, the head tube could be out of alignment with the seat tube. In other words the main triangle could be twisted.

Try the string test with the string looped round the top of the head tube or even around the handlebar stem above the headset. Check the distance of the string each side of the seat tube. Then loop the string around the bottom of the head tube and check again. If the head tube is short it may be difficult to spot a slight variation.

More often than not the problem is the front fork. Sometimes in a crash the front wheel will turn at 90 degrees to the frame and the rider will go over the handlebars. The front fork will get pushed sideways as a result.

Stand the bike against a curb and step back some distance from the bike viewing it from the front or the back. Close one eye and look to see if the front and rear wheels are in the same plane.

Unfortunately there is no simple test like the string test to check the alignment of the whole frame and fork. Unless it is obvious to the naked eye, the bike needs to go to someone with the right tools to check it.