First : When you say "the particles it filters out would sit right on its surface, pile up there, and quickly plug up the pores in it and stop the flow of air through it" don't forget that in reality, this is what also happens with commercial air filters.
Second : You say the tissue paper is weak, so is the paper in commercial filter before it was treated and protected. In that sense, we can do the same thing, by providing a support before installing the paper issue.
I personnaly first pre-install a mosquitoe screen, then add the paper towel. When the paper issue filter is no more efficient, I simply install a new one.
Simple, efficient and cost effective.