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I don't understand why it is considered a passing touchdown that accrues to the QB's stats, in cases when the receiver gains most of the yards. Consider a 5 yard pass where the receiver then runs 90 more yards. My understanding is that this is a passing touchdown, and that the QB threw a TD.
Shouldn't this be a variety of rushing TD instead of a passing TD, or at least its own thing "pass with YAC TD"?
The short answer is that it is considered a passing touchdown because it is, by definition, a touchdown that occurred on a forward pass. The receiver would still be credited with a "Receiving Touchdown" as well as the sum of the yards credited to his net total. This is simply the definition of how the respective statistics "Receiving Yards", "Passing Yards", "Receiving Touchdowns", and "Passing Touchdowns" are determined.
There are, however, different advanced statistics that reflect what you appear to be seeking, a seemingly more proper distribution of credit on such a play. A stat like "Air Yards" for example, reflects how far downfield the Quarterback's throw traveled before caught by the receiver. This, along with Yards After the Catch or YAC, can be used to more precisely illustrate how a pass play gained its yardage.
I would caution however, against giving the receiver all of the credit for yards after the catch, as often an accurately thrown ball by the QB is a big part of the reason for a long gain after the completion is made. It can be interesting to look at air yards however, for often a QB can inflate his stats simply by throwing passes that are essentially elongated handoffs to skilled receivers who turn them into big plays. If it works, it is certainly a benefit to the offense, but speaks more highly of the receivers and the scheme than it does of the QB.
First: "touchdowns scored" is a mostly meaningless statistic. A receiver who picks up 80 yards on a pass but is tackled at the one yard line has done quite a lot for his team, and while that last yard would be nice, who scores that last yard isn't all that important - the hard work was getting there.
Look at this article on predictability of statistics for QBs for example; passing TDs/game has only a weak year to year correlation (0.42), and passing TD rate (which accounts for teams that tend to pass a lot) was even lower at 0.23. Nearly every other passing statistic is better correlated; Yards/Attempt is 0.43 (double that of TD rate), and Passing Yards/Game is 0.66 (1.5 times that of TDs/game).
Touchdown is simply an ends based statistic - like Wins in baseball (or football), it measures how the player got the positive final result. As such, it's not an advanced statistic that's carefully crafted to show exactly how much someone contributed.
Measuring the QB's contribution to passing plays is a complicated work, and several groups have attempted to do so. ESPN has a helpful explainer on its QB rating statistic, QBR, which shows their approach. For passes, it looks at the expected YAC for a player after a pass of a particular type (YAC on a screen is expected to be more than YAC on a deep bomb, for example), among other things, to figure out how much credit to give the receiver for the completion.
