The advantage of having a continuation application on the "back burner" is to provide continuity with the initial priority application. This keeps any new publications from becoming prior art to an application that you are pursuing.
Sometimes it is strategic to quickly get some allowed and issued claims even if they are narrow. If there was no continuation on file before the first application/patent issued then anything later filed is subject to prior art that the original was not. Prosecuting a continuation right after, or even before, the narrow claims are issued has the advantage of getting on with the task of getting the broader claims you desire. It might involve many RCE's, an appeal, etc.
What you are asking about is not a continuation that is being actively prosecuted but one that is sitting at the USPTO deferred under 37 CFR 103(d). This must be requested before any office action and the publication can not be differed. You can request this for any amount of time up to 3 years. However the clock on the term is ticking. All the patents issued will terminate based on the initial priory date.
Reasons for a late prosecuted application might be to target claims to specific competitor's product. Of course any claims must be completely supported by the priority application as-filed. A competitors product might already fall under some issued broader claim, but a relatively narrow claim that is spot on a competitors implementation might give you a leg up in litigation or settlement negotiations.
The scenario often suggested is the company's chief counsel shows the claim to the CEO and the CEO immediately sees that they are screwed without any sophisticated claim construction discussion.
Of course there are also possibilities for divisionals, and continuation-in-part to be filed that are continuations of the continuation. To get expedited examination it needs to be requested at the time of filing, so the back-burner application can't get that treatment. But a new application that was a continuation of it could.