I would argue that it is, given the semantic shift that "to have" has undergone.

“To have” historically has had a more tangible sense of holding or owning something in a lasting, physical way. For example, in medieval and early modern English documents, “to have” frequently referred to holding physical property or goods in a manner implying true, ongoing possession. For instance, the formula “to have and to hold,” found in English property grants and other legal charters dating back to at least the 13th century, specifies that the grantee possesses the land not just in theory, but in continuing, tangible stewardship. This phrase does not simply mean ownership on paper—it affirms the right to keep and maintain the property indefinitely.

Today, "to have" is more abstract, and implies enjoying a condition or availability. In modern English we often use “to have” for intangible states, experiences, or conditions, rather than strictly physical possession. We say we “have time” or “have a headache,” meaning we experience or hold a certain condition, not that we own a concrete object. Saying “I have an idea” frames “idea” as something you possess, but it’s more about the existence of that thought rather than controlling a physical thing. We “have a meeting,” which implies an event scheduled for us to attend, not an object we keep. Over time, “to have” evolved to mark various states—emotional, temporal, conceptual—thus shedding some of its older, property-focused sense and becoming a flexible verb denoting conditions or availability.

So, because we interpret “to have” as less tied tangible possession, the original logic—that once you eat the cake, you cannot still "have it"— doesn’t strike the ear as sharply when we switch the word order.

I would suggest adding a time marker like “then” (e.g., “You cannot eat your cake and then have it, too.”) emphasizes the sequence and delineates that the action of eating precedes the attempt at possession.