UI for sorting in Files / Nautilus

How the Sort By menu appears in Files:

Elsewhere is an Icon Size chooser which looks like this, with 2 buttons inside the menu item:

Could the Sort By menu have buttons too? Eg.

Sort By
  Name    ↑   ↓
  Date    ↑   ↓
  Size    ↑   ↓
  Type    ↑   ↓

The arrows are admittedly less descriptive than A-Z, but more so than “Size”. They would function the same as the arrows on the column headings. Tooltips could indicate that the arrows mean “Smallest first”, “Oldest first”, “A-Z”, etc.

Hi! It’s an intriguing idea.

You may want to share more about the motivation and rationale.