Ligatures are handled at rendering time, partly because the terminal emulation and model layer doesn't know about fonts (it can run "headless" in a multiplexer where there are no fonts), and partly because the shaper library doesn't output information about cells but rather about which glyphs to render in which positions--that information doesn't easily map to the terminal cell model.
For extra pain, some font designers have ligatured sequences that map to a single glyph that may be several cells wide, while others use alternative glyph fragments for each component of the ligature, and others may emit two blank glyphs, followed by a triple wide glyph with a negative x offset of almost two blank glyphs in width. It's difficult to map this information to cells.
If you see the term "full width" character in old terminal related specs and documents, they are talking about 2 cell CJK glyphs/characters.
Proper ligature support requires terminals to support N bytes of input mapping to M character cells... and unlike the N:1 or N:2 cases, the N:M case doesn't allow a hard-coded list of how many cells an input string covers, it varies from font to font. And that risks software designed for one font breaking in a terminal that uses a different font, if it has a different set of ligatures, or the ligatures are different sizes.