Just adding my two cents to this debate. In my humble opinion it's pretty clear what "12" in this context means as Lilypond's syntax is always about the divisor. c4 is always a quarter of a whole note. Therefore c12 would always be a twelth of a whole note, thus a third of a quarter note. And c7 would always be a seventh of a whole note. With this in mind, why should input like "c3" yield an error if it's otherwise very consistend with the syntax and definitely unambiguous? (And the dots also don't pose problems in a mathematical sense, as it's clearly defined, that one dot prolonges the note by a half of it's value, two dots by a half and a quarter and so on.)
On the contrary side, it may lead to all the issues outlined by many of you already, like Lilypond having to define a standard grouping - which probably doesn't align with the grouping of 8th beams - or guess the grouping, and define functions to change it, and so on. Also it almost certainly would introduce new problems users might run into.
A practical point of view that also might be considered is the one of contemporary music. Just one example, because I happend to perform the piece, is the opera "Powder her face" by Thomas Adès, that uses e.g. 5/6 time signature (mostly for singular bars, with a lot of time signature changes!) - which looks very odd at beginning. But: once you're through the thought process it's far easier to read than writing "5/4" and adding the relation that "the new quarter note equals a quarter triplet of the bar before" and vice versa after a single bar of music. Things like these should be easy in Lilypond, considering it's sheer flexibility and hackability. And if I were a composer writing in 5/6, i would probably be happy if I could just write "c2 d6 e6 |".
Christian