You can get kinda close to that with OctoPrint running on a RaspberryPi. You'll still have to slice it manually, but you can upload the gcode and start the print from another browser tab.

Options have existed for awhile, but unless you have the specific slicing profile in mind (which is NOT the same for every file; some might need supports... some you might not care about quality, etc), it's not going to be one-size-fits-all.

As others have mentioned, the closet is using something like PrusaSlicer and using 'Upload and start print'. You do the slicing manually, then you just press a button and it uploads the gcode and starts printing.

There are so many variables which one might want to tweak depending on their specific use case that it seems a bit difficult to boil it down to one-click.

Especially with slicers still introducing all sorts of improvements which might need tuning of their own to adapt to older designs.

3dprinteros has an online slicer. So, assuming you are happy with the default profile and the printer is online, you can go "Upload stl" -> "next" -> "next" -> "next" -> "print".