If that was true, all a player would have to do is hold an airborne shooter to where he couldn't release the ball and had to return to the floor. Instead of giving the fouled player free throws, you'd give the defense an advantage with a throw-in out of bounds.

Ying And Yang ...
 

Make up your mind.

http://ts3.mm.bing.net/images/thumbn...9d75&index=ch1

Just Let Me Finish My Shot ...
 

It better, or else a player could get fouled in the backcourt in the act of shooting and "foul-induced" travel all the way to the basket for a layup.

OK, I get the first part.

But the second part can't be right: if the foul caused the violation, then the violation never happened.

Where is the rule on this (and by "rule" I mean rule, not case)?

The violation happened; the enforcement of it (throw in to the other team) is ignored.

Well, the question is, what causes the ball to become dead. And what are the exceptions.

In the case of a shooting foul, the applicable rules are: 6-7-1 (goal made), 6-7-6 (end of quarter, not applicable in our hypothetical), 6-7-7 (a foul other than team or player control, with exceptions), and 6-7-9 (a violation with exceptions.

The applicable exception to 6-7-7 is the try has begun. The only exceptions to 6-7-9 apply to the "opponent," or defensive violations. Further, there's nothing that stipulates the traveling rules don't apply after a foul. The only time they don't apply is noted in 6-1.

A dead ball going through the basket is meaningless.

Had this VERY recently.
Double whistle I had foul, partner had travel, ball went. I didn't see the travel, but knew the defender had bumped the ball handler. Partner didn't see bump but saw the travel. Unfortunately for the player they made a heck of a shot. We put them on the line for two, and yes I had fun explaining to the coach. The fans in attendance also thought we were mistaken in our rule interpretation.
"How can it be two shots if the shot went in?"
"Your player was in their habitual (love that word) shooting motion when fouled by the defender. The foul caused your players pivot foot to return to the floor prior to the shot, thus negating the basketb because they 'violated'"
"That was a lot of big words, so I guess you know what you are talking about, but I still don't understand it!"