I just randomly came across this question. Sorry it's so late. It is fine to model the two parts of a hurdle model separately. This paper addresses hurdle and zero-inflated models
Using data from the glmmTMB package here's an example. You could fit the data with either a zero-inflated model (
zinb below, where zeros can be from either the negative binomial or the zero-inflation) or with a hurdle model (
hnb below). Then we can see that
hnb is statistically equivalent to the combination of a binomial model for the zeros and a zero-truncated model for the positive counts.
> zinb = glmmTMB(count~spp * mined + (1|site), zi=~spp * mined, data=Salamanders, family=nbinom2)
> hnb = glmmTMB(count~spp * mined + (1|site), zi=~spp * mined, data=Salamanders, family=truncated_nbinom2)
> zeros = glmmTMB(count<.5 ~spp * mined, data=Salamanders, family=binomial)
> pos = glmmTMB(count ~spp * mined + (1|site), data=subset(Salamanders, count>0), family= truncated_nbinom2)
'log Lik.' -491.5107 (df=16)
'log Lik.' -315.2394 (df=14)
'log Lik.' -806.7501 (df=16)
'log Lik.' -806.7501 (df=30)
You can also see the coefficients if you try it.