When you say “new home” I assume you mean new to you.
Did you get a home warranty with the purchase? If so, call them.
In general, a seller must disclose latent defects which they actually know about. If they knew about this problem and did not disclose this to you, they may be liable. If the problem is with the overflow, the sellers may have indeed used the tub for baths, but never had the tub full enough that the overflow would be affected.
The home inspector has a duty to you to perform a reasonable and competent inspection (read the contract with the inspector. It may have some language regarding the scope of the inspection). Whether this duty includes inspection and testing of the overflow would be the ultimate question.
On a practical note, I have lived in my house for over 7 years now and have never used the bathtub (except perhaps to turn the faucet on to clean the dust out of it). I can’t even remember the last time I turned the faucet on in the tub. If I were selling my house and it had this problem, I would have no way of knowing about it, and therefore, no duty to disclose. You can’t disclose something you don’t know about and sellers generally have no duty to inspect.
I recently purchased a new home. We paid for a professional inspector to come in who gave the house a clean bill of health. We moved in two days ago. That same evening, exhausted from a long day’s unpacking, my spouse took the first bath in the new home. That’s when we discovered a major problem. The water from the bathtub’s overflow valve leaks into the ceiling fan fixture of the kitchen (which is the floor below the bathroom.) I kid you not - I walked into the kitchen, noticed some water on the floor, looked up, and saw that the glass bowl of the ceiling fan was 3/4 full with at least 1/2 gallon of water. We were fortunate the electricity did not blow or that a fire was not started. Clearly, we are very upset - I personally find it difficult to believe that the couple selling the home did not know this was a problem (they had lived there fifteen years and never once took a bath?) We’ve tried calling our realtor, the realtor who sold the house, the home inspector, and the people we bought the house from, but no one will return our calls, except the people we bought the house from who rather indignantly told us to leave them alone. Are we liable for this cost? If not, who is?