The rake has nothing to do with JS (which I agree is cursed, but for its own reasons, not this).
You have called a function in a way that does not give a consistent value (Date()). Such functions are hardly the preserve of JavaScript. You've failed to adequately deal with the range of values produced, with code that tries to insist that the "31st February" can be a meaningful date in February. You should accept that this is your mistake and learn to (better) avoid side effects where possible.
Also, the function isn't side effecty since it doesn't make implicit references outside its scope.
Edit responding to your edit:
Also, the function isn't side effecty since it doesn't make implicit references outside its scope.
The Date() function's output varies according to something other than its input (and even the rest of your program). Using its output without accounting for that variation means that your function, as originally written, also gives inconsistent return values, varying according to something other than its input, because it does, in fact, reference something outside the function. If it did not, the results would only depend on the monthNumber argument, and would always be consistent. I don't know what you call that, but I view it as a side effect.
As you have said, the rake is that months have different lengths, and you need to account for that. But that's not one of JavaScript's many issues.