Why probabilistic syntax for encryption schemes?
Why is the mechanism of encryption formulated in this way?


Writing $D(d,E(e,\alpha)) = \alpha$ would imply that $E(e, \alpha)$ is a unique value, and it generally is not in practice (as most encryption schemes are, in fact, nondeterministic; that is, there are a number of ciphertexts that correspond to any specific plaintext).

The text as written $\text{Pr}[D(d,E(e,\alpha)) = \alpha] = 1$ accounts of the fact that $E(e, \alpha)$ can take on a number of different values (with some probability distribution), and so matches reality (while respecting the formalism).


  I read later in the book that this definition may have the probability bound relaxed to allow for some negligible chance that decryption is not possible, and that this may be necessary for certain public-key encryption schemes.


Yes, it comes up in most lattice-based public key encryption schemes and in some code-based public key encryption schemes.  I have personally never heard of a symmetric system that had the possibility of a decryption failure; that is, where the decryption of the correctly encrypted (and unmodified) ciphertext would not result in the original plaintext.