I used to do it that way, too: chop the track into phrase-clips and use slip-edits to trim them. I did that because it's what I'd always wanted to be able to do with tape, but it was impractical until computers came along. I did a lot of things in the early days of DAW-based recording just because I could.
Sometimes, though, the transition from low-level background noise to total silence was detectable in the final mix, and I didn't like the effect. Sometimes, you'd even inadvertently lose something important, like the singer taking in a breath.
So for the past 5 years or so I've saved myself the trouble of splitting clips and instead used gain envelopes. It's always non-destructive. It's less time-consuming. It allows you to preserve as much or as little ambiance as you want, and to tweak the rate at which the levels transition to and from "silence".