Applied mathematics is funded because of its practical use. Pure mathematics, like other theoretical fields, is funded to explore ideas practically oriented people ignore. Most of the time such exploration produces nothing of practical value. But occasionally theoreticians stumble upon ideas that turn out to be fundamental to understanding something practically oriented people discover much later.
If rich people want to use their money to fund math with the goal of NOT funding math with any known useful or practical purpose, that sounds like a hobby to me.
Mathematics as a field will not cease to exist, and not everyone will be interested in (or qualified to) managing the fields progress and its applications.