An SMS app has permissions and activities used on device that Tasker doesn't create. Tasker isn't possibly used to create what you call a subset of the set of "any apps" at all.
SMS apps have Android OS-deep "SMS permissions" and Android OS has the activities, processes, and profiles for listening to the cell tower and relaying the message to the SMS app. (It's like Android has the active profile, and like the SMS app specifies the task of alerting you with a notification.
Tasker and kid apps work by using profiles and monitor services, right? An App doing so is why Android OS requires the persistent notification. I can disable my kid app on my device (on internal test on Play Store), my notification disappears, and when I relaunch the app, the service and little icon pop back up.
SMS apps rely on other intents and listeners which are already built into Android.
Here's the issue: Tasker is really sophisticated and suited to imaginative experimentation, but is not built simply as an app building environment to do what you ask here. You couldn't make a fully fledged SMS app with Tasker, even if you could rightly almost get there with enough tasks and profiles.
Perhaps... The icon could disappear and the notification could still be visible when pulled down?