Why do so many exercises in training? Why not just sing songs?

The training is part of conditioning the voice to do the right things, or the right habits, that mirror the “feel” and the world of great singing. Most singers come in with limited range, a tendency to strain or lose control, get “airy”, or carry a feeling of uncertainty and doubt through parts of their range. Singing a song over and over without isolating the “problems” or “issues” of the singer only reinforces bad habits.
“Vocalizing” or using specific exercises to train the voice will condition a singer to practice fundamentally good habits that allow more freedom of the voice without triggering the wrong muscles. As the vocalizing improves, as one eliminates bad or unhealthy habits in his approach, singing is enhanced as well. You will in essence “re-train” your voice with the right habits that will impact your singing.