Why do we add "αd" to N in Laplace Smoothing?
You're right, the idea is that, if $\alpha d$ is not used, then you might have $\hat{\theta_i}$ bigger than 1.
That being said, I don't think the main reason of Laplace smoothing is to have probabilities greater than 0. I think the important thing is that it acts as a regularization technique (a smoother, indeed).
The smoother works the following way: we are mixing the empirical observations ($\frac{x_i}{N}$) and the theoretical distribution without data ($\frac{1}{d}$), and $\alpha$ is a parameter that controls how much do we care about the theoretical distribution. The idea is: the more we trust the theoretical distribution, the more regularization we are adding. It even has a Bayesian interpretation in terms of priors and posteriors.
So I don't think it has to do with probabilities being between 0 and 1, I think it has to do with performing the mixture of distributions (or update of the posterior, if you want) properly.