Range is expressed in feet.
Feet is a measurement of distance.
Therefor, Range is a distance.
QED
Your logic is flawed.
Range
does not equal feet.
Range is [often]
expressed in feet.
Sometimes it is expressed in squares.
Even if given in feet, it can still have different meaning.
Sometimes it is given as increments.
Sometimes it is given as touch, short, medium, long.
Sometimes the range given is a hard max.
Sometimes the range given is a point where you don't accrue range penalties.
Question: So in which case is 'range' the same as distance?
Answer: 'Range' can have different arbitrary meanings depending on what type of action is being discussed. It can also have different arbitrary designations depending on the level, weapon, or other factors. Thus, range does not describe a real-world number. Range describes an arbitrary limit imposed on various abilities, and does not use real-world math. If you are told to double the range, then multiples are added, not multiplied.
In other words, 10 feet always means 10 feet.
However, a 10' range can have several different meanings.
Thus, range does not equal distance, even though range refers to distance.
QED
Caveat: If a feat (or class ability, or whatever) said something like: your bow can fire twice as many feet, or double the distance, then you would use real-world math.