Could a straight line demand curve ever have the same elasticity on all its points?

No, a straight line demand curve can not have same elasticity on all its points. This can be explained as follows.

Elasticity of demand is defined as the ratio of percentage change in quantity demanded and percentage change in price.The mathematical expression for the elasticity of demand is given by (dQ/dP)* (Q/P). Along a linear demand curve, dQ/dP is constant but Q/P keeps on changing.

Now, consider a linear demand curve given by P = a - bQ------------(1)

Hence, The inverse demand is given by Q = a/b - P/b-----------(2)

Differentiating (2) with respect to Q, we get dQ/dP = -1/b---------(3)

Putting the value of (3) in (1), we get the elasticity equals to (-1/b)*(Q/P)---------(4)

Hence, it is clear from (4) that elasticity is not the same along a linear demand curve because quantity and price keeps on changing.