Maybe some teams operate that way. But wouldn't this also be a function of the IR rule? For example, if the Pats think there is a remote chance Andrews could play this year, he would need to make the initial 53 man roster and then be placed on IR. If they IR him immediately at cutdowns, he'd be ineligible to return at all this season. To choose this route, it means they must cut someone that they might intend to have on their week 1 roster. Players who are not "vested veterans" have to go through waivers, but vested veterans do not - they are free to sign with anyone.
Wouldn't it make more sense to tap someone like Brandon Bolden on the shoulders and say, we're temporarily cutting you for roster space, but we will immediately resign you if you're ok with that (and obtaining Bolden's buy-in)? As opposed to cutting someone like Keion Crosson and hoping he makes it through waivers unclaimed before you can resign him?