Thanks for the support! I agree, it's better to have many communication methods.
The thing is, I have email, push and SMS all turned on. But I don't get any of them for a new inquiry. They work fine for the second message, or reminder to write a review, or offer expired.
It's a weird problem and I wonder if anyone has a workaround. Maybe a service like Zapier or IFTTT could connect with Airbnb and send me more reliable notifications?