When exporting, there should be an option ,“Export to multiple images”, that allows you to choose a grid size (eg 8x8) and export the sprite as many 8x8 images, so a 32x32 sheet of 16 sprites exports them all individually.
You can already do this with slices. There’s no auto-slice tool, but if you enable snapping to the grid, it’s quite quick to make a grid of slices. Unfortunately, the separate-image export is currently command line only. You can find the documentation here.
You could also author your work not as a sheet, but as frames in an animation, and export those as separate files. If you select a non-animated file format, such as PNG, in the Export menu, Aseprite will give you the option to export as individual frame files.
If you need to work as a spritesheet but export as individual images a lot, then Pyxel Edit might be more appropriate, as its workflow is built around grid-based spritesheets and tiles, whereas Aseprite’s more geared towards streamlining animation and doesn’t directly deal with spritesheets before export time.