In case of public universities, the mathematicians would ultimately have to convince the public, but of course in practice that means politicians and national grant agencies, who indeed report to politicians and the politicians chase popularity to the extent needed for reelection, so public sentiment has some relevance.
Generally when you come to someone holding out your hand asking for money, being humble is a good first step, instead of getting angry at people for not sufficiently appreciating the big myterious value of what they are doing. Explain the spiritual value of the pursuit, they may or may not be convinced. This may be called a deficiency of democracy if your aesthetic values are different, but typically people don't like to fund abstract disconnected things, like expensive space travel unless it's embedded in some tribal jockeying like during the Cold War. In normal times people just want research that has life enhancing practical outputs. Nevertheless currently the West is structured as electoral democracy, so convincing the plebs is indeed necessary, no matter how lowly that feels from the ivory tower.
what about people who study birds, or say, olms? people want that kind of research? the people who are doing it surely do. astronomy and eg star watching in deep space? i could go on for days
people like doing mathematics, similarly to studying endemic organisms or other stuff of no direct utilitarian value
how many parents are happy to pay the fraction of taxes that go into something that their children might want to choose as a calling and fall in love with?
it’s okay to discuss whether something should be publicly funded, using “people want x kind of research” as a supposedly obvious or common sense argument is fallacious in the sense that it implies an idealised steelman model of the general public. that model is divorced from reality