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A recent question made me think of this peculiarity (in my eyes at least) of american football.
I get the logic behind crediting a TD to both the receiver and the QB. But the QB gets credited for the yards a receiver takes after completing a pass, but he's hardly responsible for the athletic ability and tenacity of the receiver rather than the QB.
Is there any rationale behind this?
Largely, because of how sports statistics were created.
For a long time, statistics tended to be results oriented. Football statistics came from play by play, which went something like (below is my approximation, not actual PBP):
SF 20, 1st and 10: Incomplete Pass to 84 Rice
SF 20, 2nd and 10: Pass to 84 Rice 18 yards to SF 38
SF 38, 1st and 10: Craig, 5 yards to SF 43
Etc.; it gave basic information, but not the detailed information you have now. You can see an example of this on Pro-Football-Reference's PBP of the 1989 Superbowl.
As such, you can't separate things like run-after-catch and things like that from the total play; you simply get a single number.
With the advent of advanced statistics, you get interest in having that additional detail - as you say, passing yards with little run after catch are very different from ones with more run after catch.
In the interest of keeping things comparable from one era to the other, "old" stats will tend to stay the same: QB Passing Yards will still be total yards, for example. But instead, you have new statistics.
Advanced statistics often try to strike a balance between simplicity and descriptive power, which is likely why ANY/A doesn't include YAC adjustments. Additionally, any statistic that is not determinable from play by play will not be possible to get for older games, and may be difficult to get for games before it was created; Football Outsiders' Game Charting Project for example requires multiple volunteers watching game footage of every game ever played, have been for several years now, and are only to about 1989 currently.
As with many common sports statistics, there is some "arbitrary-ness" to how certain things are defined and many stats are simply kept around by convention because they keep things simple and are easily understood by fans. In American football it's incredibly simplistic to just split every yard gained into either rushing or passing--and quarterbacks get credit for how well the passing game goes. For fans with a more nuanced view of how things work, it's up to them to determine how useful the stat is and make their own conclusions.
Should a running back not be given as much credit when he is going against a horrible defense? Maybe not, but the standard statistic of rushing yards is still kept for comprehensiveness.
A better analogy for the quarterback/passing yards situation might be the statistic of assists, whether in basketball, hockey, soccer, or other places: the player gets credit for an assist based partially on how their teammates perform. The assist stat for a great passer in hockey can fluctuate based on many things. However, the fact that a great passer should get some credit for creating offense is somewhat synonymous to how a quarterback sometimes creates yards after the catch with their own good play.
Consider two quarterbacks who complete the same number of passes over the same number of yards (with yards after catch taken removed). One of the two may be able to use skill to lead his team to more yards after the catch and thus more passing yards. This can be accomplished by finding the right receiver who is "more open" when he could otherwise have thrown to another open receiver, or by properly leading the receiver (i.e, the receiver doesn't have to jump, dive, or stop to catch the ball), or by having exceptional timing to spring the big play, or by using strategies or methods that can confuse the defense and help the receiver have a more open running lane after completion.
That's not to say that every quarterback deserves every passing yard that they are given credit for. I mean, some QB's end up with many more yards simply because their team passes so often compared to other teams. But they can and do have at least some impact on the yards gained after reception.
