I do not understand why however is used as the end of this sentence.
Ending with "however" like that would usually be "setting up" for a sentence to follow. That sentence would provide a contrast, saying what the most common approach would be:
- "As we will see below, there are some cognitivists who do not take a Gricean approach, offering a semantic explanation instead. This is not the most common approach, however. Much more common would be to copy and paste explanations from StackExchange."
Since you stopped your example after the "however", we don't know if there's such a next section or not. But I'd feel it would be weird without it, sort of dangling. "however... what?"
Could I front however to the head of this sentence
(Sidenote: using front as a verb like this is not "correct", though it can be understood as a casual and expedient way of "verbing" a noun--people might do it, consciously knowing it's not correct English. When people use "front" as a verb it usually means something else... as short for "confront". So here you probably want to say something like "Could I move 'however' to the front of this sentence?")
Why does this author like to end a sentence with 'however', or even 'instead'?
Mechanical questions are easier to answer than "why". Why don't they give their example first, and then say "as we have seen" instead of giving us a preview with "as we will see below"? (Why write something that sounds so boring in the first place? :-P)
The positioning of these words has to do with guiding people along a train of thought. You're being told something first...but then being asked to "change gears" to realize the thing you've been told might not be the most important. Good placement of these pauses and turning points helps keep readers on track with the points in the order you want to make them.
The reason for choosing any given ordering and those contrasts may be fully reasoned--or it could just be the order that the author happened to think of them in. Where you put the howevers has more to do with how the setups sound "in your head" than anything.