The sad thing with the Avenger is that of the majority of the pulp heroes, he's actually the one most suited for superhero style treatment in comics or movies.
Doc Savage: suited mainly for international and large threats. Don't often see him going after gangsters or solving a murder mystery or just a bizarre crime.
Shadow: whereas the Shadow was always focused on the criminals and crimes. Some were bizarre or large threats, but rarely did he cross over into Doc's territory. The Shadow also has the drawback that he is best a semi background or supporting character in his books. His influence should always be felt, but you really don't get to know the man. Hard to pull that off consistently in comics.
Secret Agent 'X': cool stories and a variety of types of crimes and threats that he can face, but like the Shadow, a big part of his thing is the character himself is a cypher. This also poses the problem in a visual medium like comics, his whole thing is to stay in disguises.
Phantom Detective: He has a real identity, tackles a variety of crimes, but his thing again is disguises which doesn't really play to the character's strength in visually based media.
The Spider: Real identity and goes after some of the worse super-villains imaginable. AND he has a couple of strong visuals to go by (whether interpreting the literalness of the pulp stories, or the slightly watered down looks on the pulp covers). The downside here is that outside of the original stories, no one has been able to capture the passion of the characters and instead play up the violence and thus instill a psychopathic nature that's not really there. Look at the fate of all Dick Tracy's villains, but he's not played up as a homicidal cop.
In terms of adapting to a visual medium, especially comics, the Avenger has the strengths of the others, but not their drawbacks. He fights international crime and world threats, but also the gangsters and murderers. His origins are based in crime and wiping it out (indeed his origin and back story are easily as powerful and tragic as those that fuel Batman and Spider-man). In that, we know the man and his friends. They are more fleshed out than that of the Shadow's. Just as he is more approachable a character than either the Shadow or Doc, and more can be done with him. While he is now a bit apart from society, he has a whole past life that can be explored. He has the passion of the Spider but not the cold-blooded killer. He has a unique look that fits well with the comics (especially if following DC's cue and using the paperback reprints as a model).
The Avenger could easily be of Batman caliber in the comics if played to his strengths whereas in the pulps he came off as what he was, a pale imitation of Doc & the Shadow. This comic instead made him even more bland.
One of my favorite single issue comics is actually the 1970s Shadow story that guest-starred the Avenger.
