I agree with you that errors are useful feedback for coders who don't know the ins and outs of an API. And every programmer is in that group at some point. But the difficulty in identifying this particular bug doesn't stem from the API decisions.
Whether Dates throw an error, or work with what they're given, has no bearing on the subtlety of this bug. Either way, tests that don't replace Date will fail to identify it most of the time, and tests that do, based on its use within the function, would be called wrong-headed by many.
Either way, the bug only shows up at the end of months longer than the target month, and that infrequency has nothing to do with the peculiar design choices of the Date API. It stems exclusively from the evaluation of Date() called with no arguments returning different values at different times—behaviour you have not objected to, and which I'd expect to be considered entirely appropriate, in fact its very point—combined with an attempt to use that value, whatever it may be, without due consideration.
Since the month is the only part of interest, there's no reason to allow the other parts to vary at all. Fixing them, as I suggested at the beginning of all this, is the simplest approach, but setting them first, as has also been suggested, would work too.
You can once again complain about JS design decisions and I'll agree about many of them, but, as much as you might like it to be, and as annoying as so many of us think they often are, here it is beside the point. The perniciousness of this particular bug stems from unnecessarily calling a function with inconsistent output and then improperly processing that, instead of using a function call with always-predictable output.
I've tried to point that out in all the ways I can think of, so if it's still not getting through, I give up. And if your acknowledgement was too subtle for my sleepy brain, and I've ended up overexplaining, then I'm sorry.