Not gonna argue Bilic expertise, he is solid manager but nothing special. For me the drama around his sacking is related to the fact that he managed to get promotion from one of the toughest league, that gets WBA 200 million pounds and they don't buy any new players and expect him to have great results in PL.
Than they hire new manager, he looses all the games except Liverpool one, with big handicaps and just says he can not work with limited team like this and that they need to buy new quality players to avoid relegation. Well no shit Sherlock, that is just the point, Bilic was doing the best possible thing with players he had.
He didn't do great even in the Championship - he did well enough to keep his job but nothing more. If we look at the bookies odds (which is overall the best possible 'average' to take), WBA were 2nd after Leeds to be promoted so them going to PL wasn't an overachievement at all.
I agree that he should have been given some money to spend, but I'll also like to add that statistically, Allardyce and his style of football will bring more points with such a crap team than Bilić and his style would.
Allardyce will play the long ball game and will base his game on set-pieces, whereas Bilić wants to play football - at least tries to.
Bottom line - Klopp and Guardiola combined probably would not be able to save WBA this season. That said - with this team - I'd give more chance to Allardyce surviving than anybody else - although he has 0.5% and others would have 0.3%.