Is saying that the demand is elastic or inelastic too oversimplified? As we know that elasticity varies from infinity to zero as we move along a linear demand curve, then is it correct to label any demand curve elastic or inelastic as a whole?
From your Q I am not sure if you are asking solely for linear demand functions or demand functions in general.
When it comes to demand functions just in general  actually, we do not a priori know that &quot;elasticity varies from infinity to zero as we move along a linear demand curve&quot;  there is a class of constant elasticity demand functions for which price elasticity of demand is always the same. For example:
$$P  = Q^{1/\epsilon}\implies Q= P^\epsilon$$
will have constant price elasticity. Since price elasticity is given as:
$$e_p= \frac{dQ}{dP} \frac{P}{Q}$$
In the above case the elasticity would be simply equal to $\epsilon$, as:
$$\frac{dQ}{dP} \frac{P}{Q} = \epsilon P^{\epsilon-1}\frac{P}{P^{\epsilon}}= \epsilon$$
The constant price elasticity functions are relatively popular in the literature and for these functions it clearly makes sense to label them as elastic or inelastic based on value of $\epsilon$.
However, even for a linear demand functions where price elasticity of demand changes along the function it would be reasonable to say that one demand curve is more inelastic relatively to some other demand curve, respectively being more or less elastic over some range. In such situations however the terminology might often be abused and demand function simply termed 'inelastic' - this happens especially in many undergraduate texts and I suppose you could call it an 'oversimplification'.