Before we come down too hard on movie faking, some consideration should be paid to just how the playing is being added to the film.
For a Hollywood style production, one that's meant to look good as well as sound good, the "faking" is a necessary part of the process. When the film is shot, the sound that you hear with the scene is not directly recorded as part of the shot. instead, the scene is shot as is and then put "in the can".
Later on, the artists who do the actual playing record the sound to tempo, and then it is "stripped in" (and don't ask me where that term comes from) as part of the post-production process.
The reasons behind this become obvious once you start thinking about them:
1) Recording equipment restrictions - no mikes or booms or cables to clutter up the scene visually
2) Error issues - re-recording a musical track to get it right is cheaper by an order of ten over redoing a filmed scene for the same purposes
3) Continuity issues - most films consist of a series of different shots in each scene, and recording the music at the time would cause all sorts of problems when trying to match the various shots up for the finished operation. Normal tempo variations alone would play havoc with "doing it live".
4) Skills issues - as stated above, not all actors have any musical ability. A good part of the time, the actors are acting and the musicians are playing, with the tracks being brought together in post-production
I used to have a friend who did a lot of film work out West, this being the recording of the music for a film (not the acting). I once mentioned to him that it must be fun to do all that playing, and he answered that it was the most boring process in the world. Trying to match tempos (not just the "official" tempo set at the start of a piece, but also when there are inadvertent speed up and slow down issues) made it all very tedious.
Some examples here, a bit more expanded than those above:
Tony Curtis for starters. He has been in not one but two "saxophone centric" movies (the Billy Wilder classic Some Like It Hot, and the not so well known (but much more musically oriented) The Rat Race). In both movies, he spends a good part of his time on screen with a sax in his hand (tenor in SLIH, tenor and alto in TRR). In both cases, he seems to be moving his hands "correctly" for the most part as far as key action is concerned, but "out of sync" with the film overall.
This would seem to be indicative of poor (or no technique) on his part, but I would invite you to take a look at one of his "co-stars" in TRR, some guy by the name of Jerry Mulligan.
Watching Mulligan's fingers move on his horn reveals the same "right technique, wrong timing" issues as are present with Curtis. I'd say that most here would feel that Jerry Mulligan would be getting it as right as he could. Thus, the answer lies not in the actor's ability but rather in the limitations of the medium.
In SLIH, the music is a bit more peripheral to the story than in the other film. However, compared to Lennon's antics on the fingerboard of the doghouse bass, Curtis seems to be doing pretty well, if somewhat out of sync. Once again, the marriage of the film and sound seems to be the cause.
In some films, you can actually hear a difference in quality/acoustics/whatever when they switch from the sound recorded during a scene to the post-production recorded stuff. However, that distinction has gone away since the 1970's or so. Better equipment and effect processing, I would imagine.
I have seen mention of saxophone playing ability in a Tony Curtis bio, but he did not claim to play the horn well. It is odd that he appeared in two different films which both featured the crooked horn, though.
Also regarding the sax, there is a movie with Danny Kaye as a eggheaded music professor who is engaged in compiling an encyclopedia of music during the 1940s. Various musical talents from the era appear in this, and when Charlie Barnett is on screen playing, again the finger action does not directly match the sound coming at you from out of the speakers.
With both Barnett and Mulligan, clearly masters of the instrument, having these "mismatch problems", it about sets the seal on the problem being with the process, not with the players.
Regarding Steve Allen, a true genius in many fields, it is true that he was no Benny Goodman. (For one thing, he wasn't an a-hole in real life, unlike the great Goodman - I once spent an hour and a half on a plane next to Steverino, and was truly amazed by his depth and good humor in the bargain.) However, he was able to play clarinet, as demonstrated by his performance on same on his television show in the mid-1960's. Benny Goodman good, no. But, still able to get around on the horn.
(The most remarkable thing about Allen was that he had more published tunes to his credit than anyone else at the time, even exceeding Irving Berlin's output. And, he did it all without being able to read a single note of music.)
The Benny Goodman Story is remarkable in one other area - the most blatant musical instrument plug ever put on film. When the young Benny arrives at his ancient teacher's house for a lesson, another student (either also arriving or just departing) notices that Benny has a new horn:
Other Student: "Oh, a new clarinet, huh?"
Other Student: "Selmer?"
Old Vito up in Wisconsin must have gritted his teeth in anger when that happened.
A friend of mine was the tenor sax player in the Royal Tank Regiment uniform during the "Portobello Road" scene in Disney's Bedknobs and Broomsticks, this back in his college days at UCLA. He went to some trouble to learn the chart for the scene, and it was played by the on-screen crowd when it was being filmed, but (once again) the final version (recorded by other musicians) was all stripped into place post-production. To shoot the one scene, he was at the studio for almost a week, and spent plenty of time eating well at Disney's expense.
However, Jim was lucky enough to have about fifteen seconds of his on-camera time used in the release version (and slightly more in the one released on video), plus he managed to make the "lobby card" (a small photographic scene that was hung in movie theater lobbies to plug upcoming films) in the bargain (albeit only from the back side).
He has one of them hanging framed on the wall at his house, and I took the trouble to get him listed in the ImdB, this to further insure his immortality. However, I was the only person who ever identified the uniform he was wearing. More useless trivia...