Instead of customizing the help content (more about that in a bit), let's remove the page about private beta from the help center entirely and instead make a canned, featured meta post at the start of private beta. (It can be post #1.) Featured posts stay featured for a month, which is long enough to cover private beta, and they appear in the Community Bulletin section, making this advice more visible to new users of a new site. No manual intervention would be necessary to remove a help topic after a site leaves private beta (or if it never had a private beta to begin with); this would be fire-and-forget at the beginning of private beta without affecting anything else.
If we want to keep the topic in the Help Center I can see a couple possibilities, but they involve more work. One approach is to use per-site topics, like how to create a minimum, complete, verifiable example on Stack Overflow. (They have it, so it's possible.) This is probably a per-site setting, not a switch that can be flipped for "all betas", but this special how-to-ask topic is primarily relevant during private beta, not the years that follow, and new private betas aren't that frequent (especially now), so doing whatever they did on SO to add that topic doesn't seem like it would be too hard.
In looking at the Help Center, though, I noticed that we also have how do I ask a good question?, which is general and not specific to betas. It also covers things that are mostly not yet relevant during private beta -- for example, it starts with advice to search first and then talks about scope (as defined in the help center). A private beta doesn't yet have enough content for meaningful search, nor does it have scope defined in the help center. So maybe, instead of having a second topic (a) at all and (b) switched on and then off for private beta, we could have a single topic about asking questions and change its content when a site crosses into public beta. The Help Center should always contain "how do I ask a good question?", and during the private beta it should give specific guidance for private beta.
There are ways we could make this better in the help center, but they would require manual intervention at launch and then again at the beginning of public beta. An automatic featured meta post seems both easier to manage and more visible, so maybe we should just pull this topic from the Help Center and do that instead.
In the time since I wrote this answer, the Area 51 process has changed in a way that produces very few new beta sites. This means it's even easier to do a custom solution for any new private betas (I still think the featured meta post is the cleanest). It also means that it's even more unbalancing to have this topic in the help on every site; new betas are no longer the common occurrence they once were. A common response to requests to add some detail to the help is the help is supposed to be general not exhaustive. Let's make that true here too.