Shaw does this rather clumsily, but taking all in all, it's not entirely his fault. He came out of the tradition of melodrama, and was one of the first to pull away from it. Pioneers don't have the advantages of knowing the tricks of those who come later. They are the inventors of the tricks. If we compare GBS to those who came after he is certainly cartoonish. Yet compared to those who came before he shows a more realistic character. Still, it is worth noting that an artist who deserves credit from scholars may not be worthy of the effort of a production now. It was a favorite theme among my theater friends in the 1970's that Sarte had stolen everything from Artaud, and it's sorta true. Nonetheless, Sarte did it better and remains at least watchable/readable. Artaud is not that interesting anymore.
As for Shaw, John Osborne thought him a complete fraud. I don't know that this is true, but I am figuring that Osborne has some credibility here. More than, say, me. Or you. Others disagree, and I suppose they have more credibility than me also. It's fascinating to read the Wikipedia article and realise that the collective critics pretty much boil down to one critic who has successfully fought off the others. In this case, whoever controls the Wikipedia legacy of Shaw has decided that Fred S Crawford's opinion is the bee's knees. Crawford insists that everyone who criticises Shaw was nonetheless influenced by him, no matter how far he has to stretch to illustrate that. Everyone owes everything to Shaw, it st seems. Coward, Ayckbourn, Stoppard, and all the absurdists. Even Osborne who disliked him. Crawford finds it not only significant, but definitive, that the cahttering classes talking about Shaw gave birth to the word shavian, which is still in use today. Except it isn't.
Welcome to the petty world of artistic criticism. What remains is that we can examine Shaw for his ideas and see how those have held up. I'll give that summary to George Orwell, who makes a genrral observation that is also quite good, then narrows it to Shaw.
As it happens, George Orwell in his 1946 pamphlet James Burnham and the Managerial Revolution does shed light on the Glenn Beckish claim that Shaw’s dual embrace of communism and fascism was broadly typical of Fabians or other sorts of socialists:
English writers who consider Communism and Fascism to be the same thing invariably hold that both are monstrous evils which must be fought to the death; on the other hand, any Englishman who believes Communism and Fascism to be opposites will feel that he ought to side with one or the other. The only exception I am able to think of is Bernard Shaw, who, for some years at any rate, declared Communism and Fascism to be much the same thing, and was in favour of both of them.I have complained about the teaching of Shakespeare as well. He's next. If you want to get a head start, read Polonius's speech to Laertes.