Yes, and I've said that I agree with that in general. I know that this isn't hypothetical; that's exactly why I keep saying that throwing an error doesn't help you find this bug early at all.
Even the silent weirdness can be caught by the most basic of tests checking output against input, but only if your function works the same way on every invocation.
Whether making a giant fuss (as you'd prefer) or making the best of it (as it actually does), the setMonth method always works the same way. My code always works the same way. The setDate suggestion makes the code always work the same way.
Code that always works the same way is easy to test.
If the day of the month is constant and incompatible with setMonth, whether there's a thrown error or just an unwanted return value, a simple test will reveal that on every test run. If the day of the month is constant and always compatible with setMonth, the test will pass appropriately on every test run.
The bug in the code you originally presented comes from working differently over time. That's why, most days, tests won't identify the problem, even with a fussy, noisy API. Most testing days, the date will just happen to be compatible, and even the fussiest, noisiest API will carry on without any mention of the problem.
The reason the original code works differently over time has nothing to do with the silent, unexpected behaviour of setMonth. It's entirely down to calling Date() without arguments, the entire point of which is to give different values over time. That call effectively introduces state that is not controlled by the function. And not bringing it under control is the real source of the bug.
Yes, absolutely, JavaScript sucks. Make F# the only supported web scripting language! But JavaScript's suckiness is not the cause of this particular bug. JavaScript's suckiness is not the reason this bug is hard to catch. The real problem lies in code that functions differently over time when it should (and could easily) be consistent. That's what actually makes it hard to test.
Plenty of other languages and API design choices still allow code that functions that work differently over time, which is why, as justifiable as the complaints are in general, those factors are irrelevant for this particular bug. Write code that always works the same way and the problem goes away. That's the real core of the issue.
Obviously, that's easier said than done, and it's irritating that neither loud errors nor most testing will help you in this regard, but that's the way it is.