Binomial logistic regression for multiclass problems

Applying the independence of irrelevant alternatives (IIA) is an inherent assumption in multinomial logistic regression but not binomial. Is it therefore okay and possibly better to use $K$ binomial logistic regression models instead of multinomial logistic regression when handing a multiclass classification problem with $K$ classes? Is this approach typically employed or not for any reasons in any scenarios (e.g. predicting elections)?

The primary concern I can think of is interpretation of probabilities. The $K$ binomial logistic regression models would allow one to understand the relative probabilities of class $A$ versus not class $A$, but would not have easy comparison of the probability of class $A$ versus class $B$. To compare the probabilities, some kind of alternative method (e.g. apply softmax function) would have to be employed, but perhaps this is just as misguided as the IIA assumption.