Why the Desktop always takes the hit first
Cmd+Shift+3/4/5 default to saving on the Desktop — that's been true since the day macOS was installed, and most people never change it. The system never nudges you with "you've got too many screenshots, time to clean up," and there's no auto-archiving. Shots just keep piling up until you decide, on your own, to do something about it.
Two old fixes, each with a cost
One option is periodically dragging Desktop screenshots into date-named folders by hand — the catch is that it only happens if you remember to do it, and after a few weeks of putting it off, the backlog gets big enough that you don't want to start. The other is redirecting the default save location away from the Desktop via the "Options" menu in Cmd+Shift+5. The Desktop looks clean, but the screenshots are still piling up exactly the same way, just somewhere out of sight — and finding one still means wading through a folder of timestamp-named files. Both solve "out of sight," not "findable" — see built-in macOS screenshots vs. a screenshot tool that organizes itself for the fuller comparison.
How far the built-in "Stacks" feature gets you
Right-click the Desktop and macOS offers "Use Stacks," which auto-groups icons by kind, date, or tag — click a stack open to see everything inside it. Used to tidy up a Desktop covered in screenshots, it works instantly: dozens of icons collapse down to a handful of stacks. But Stacks only changes the visual presentation — it's still grouping the same files by an external property ("these are all images," "these are all from today"), not by what's actually inside them. It doesn't read the image, doesn't title anything, and finding one specific screenshot still means opening the stack and scrolling through thumbnails one by one. It solves the same layer of problem as the two fixes above: whether things look tidy, not whether you can find them.
A different angle: not cleaning up, never landing there in the first place
Shotbox uses the same hotkeys to capture, but the image never passes through the Desktop — instead it goes straight through recognition, titling, and indexing into its own library, so the Desktop never gets the chance to pile up in the first place. The confirmation strip that appears the instant you capture has a "discard" button right there — an accidental or unneeded shot gets dismissed on the spot instead of quietly taking up space, so what's left in the library doesn't balloon into a new graveyard either.
What about the existing backlog
Here's a boundary worth stating plainly: Shotbox only handles new screenshots taken through its hotkey — the historical pile already sitting on your Desktop won't be automatically pulled into its index or cleaned up. Dealing with those old files is still on you. The practical move is to set aside an afternoon, delete what's obviously junk, manually move anything worth keeping into Shotbox or a fixed archive folder, and let automation take over from there so the Desktop never builds back up.