We Americans tend to be lazy speakers. We barely open our jaws and use our tongues to enunciate when speaking. That does not translate well to beautiful singing. In order to make beautiful vowel molds, the jaw must be down in the "two finger" position which makes it harder for the tongue work to create the proper vowel molds. We also tend to speak without engaging our abdominal wall muscles in order to project. We also forget to breathe deeply enough to have the power to support a tone through entire phrases.
Beautiful singing absolutely requires attention to good breathing, proper support, and utilizing the teeth, tongue and jaw to make the diction intelligible. But what happens between the consonants is what is so important to beautiful singing. The tone line should be disrupted for as little time as possible. The consonants should only break up the line briefly. What tends to happen unless the singer is paying attention, is that they take too long forming and executing the consonants. It could be that there is an inordinate string of consonants to get through or that some consonants are just hard to produce quickly. Some offending consonants are "r" and "l" and "w" which tend to close the jaw, particularly when they are combined such as in the words "world," "twirl," "wrong," "our" or "hour" or anything like unto them.
Then there are the consonant and vowel combinations that require the tongue to do calisthenics. Strings of letters sung at a fast clip combining tongue flips, both voiced and unvoiced consonants and barely one singable vowel in the mix can be challenging for the singer. We have a line in one of the choruses from Messiah we are singing right now that goes "Great was the company of the preachers." Try saying that five times quickly in succession at a very fast clip in rhythm! Don't forget to flip the "r's"! Getting through the "gr," "tw," "vth" and the "pr" quick enough is rather challenging, let alone allowing enough actual singing time on the few vowels in that string of consonants!
The shaping of the phrase happens during the tone line not during the disruption of the consonants! Careful management and timing of the consonant strings to happen as briefly as possible is all important for beautiful singing. When the conductor says "sing from consonant to consonant" he asks you to support the vowel tone for as long as possible between the consonants. He wants as little disruption of the line as possible.