Is there something shallow about "wordplay?" ––Perhaps. I find it interesting that you "rarely find wordplay particularly deep," and you "can't readily think of an example of wordplay that could be plausibly described as "deep", or even "deep-looking." Of course, it is interesting because you presume that philosophical depth is not superficial.
There are numerous grammatical jokes: I would look into some of Lewis Carroll's writing both for their quality and Wittgenstein's familiarity with his writing––Chapter 6 of Through the Looking Glass features a conversation between Humpty Dumpty and Alice; it's cute, and it addresses the question of meaning, in a humorous manner. A professor of mine once told me that one of Wittgenstein's favorite grammatical jokes was the following.
What's the difference between a hairdresser and a sculpter?
––One curls-up and dyes, the other makes faces and busts.
Now, it's not the funniest of jokes (or even grammatical jokes): for one, I'm unsure if there were even sexual connotations of "makes faces and busts" when Wittgenstein was alive. There are others as well: the famous "eats shoots and leaves" panda joke, numerous accidental newspaper headlines, mistranslation and so on. For example, "firefighter helps burn victim" is funny(ish) because of worldplay.
Perhaps, it is the familiar meaning of the phrase or joke coupled with another, often, ridiculous meaning that lends the phrase or joke its particular sense of depth. What gives a phrase its familiar meaning (by familiar, I mean something like common-sense or obvious) is the context, in which it appears––or the circumstances, in which it is said. Or, to put it another way, it is the familiarity of the context/circumstances when a word/phrase/sentence is used that gives meaning to the phrase/sentence. In unfamiliar circumstances, or if the phrase occurs without a context (as jokes, esp. grammatical jokes, seem to do), then the meaning may seem to be hidden. The meaning, it is said, is beneath the surface. (Apparent Depth.) But if we know, for example, that firefighters don't tend to hurt people, headline writers are liable to make mistakes, and so on then familiar meaning is clear––even if the grammar points to the possibility of another interpretation.
The problem is that we tend to think of philosophical issues as deep. Yet, we use those same sentence constructions, which contain words like "being" and "know," in everyday language. Extraordinarily, in ordinary circumstances, we are seldom dumbstruck when someone says, "I know that such-and-such is the case." Yet, when doing philosophy, "what we know" and "what Knowledge is" are apparently very deep questions.
But it is important to keep in mind that it “is only in normal cases that the use of a word is clearly laid out in advance for [people]; [they] know, are in no doubt, what [they] have to say in this or that case” (PI, §142). And further, the “forms of language have the character of depth,” and this character of depth produces false appearances and misuses of words (PI, §111). For words like "betting" and "telling," we are mistaken to think of them as "of the same kind." Wittgenstein would say that they have the same "surface grammar," yet differ in their "depth grammar."
("To bet" is to utter "I bet...." It is a speech act like "to promise, "to bequeath," and so on . "To tell" is not.––If interested in this look into J.L. Austin's How to Do Things with Words and John Searle's "Speech Acts.")
These are all reasons why a strict grammatical analysis of a sentence will not reveal its meaning. For example, the question "can you help me with this" is often not a question; it is much more often the expression of a wish or a command. These sorts of problems always arise when non-native speakers converse with native speakers. (But these problems should also point to obstacles in translation––esp. from dead languages!)
Anyway, there's much more that could be said about the issues surrounding grammatical jokes. But if you're interested in Wittgenstein and these sorts of issues, I'd recommend John Verdi's book Fat Wednesday: Wittgenstein on Aspects. It is quite clear and clever.