There has actually been some revival in multi-level logic recently. [...] Sure, the gates become more complicated, but I don't think that is such a big issue
Multi-level logic is simple to implement for storage and communications, but the complexity in computational logic (number of possible operators) blows through the roof. The possible configuration space for even just a handful of interconnected gates gets too large to brute-force through, and there has not been much research in the last few decades into multivalue logic (as a subfield of mathematics).
For
N-level logic, there are
NN unary and
NN2 binary operators. For 4-level logic, that is 256 unary operators, and 4,294,967,296 binary operators. Depending on the gate types, there are usually more possible gate configurations than the operators, although a tiny subset of the operators is needed for Boolean/binary algebra. The issue is optimization: even with a handful of operators, the space of possible configurations is too wide to effectively search.
So, I would claim that it is precisely the complexity that is the issue.
Someone would have to see what the electrical components for multi-level logic gates are, then do
inefficient multi-level logic gates, then combine a few to implement the standard operators for binary algebra, then implement some kind of search across the huge possible operator space to look for efficient configurations (perhaps do it in reverse, looking at what kind of operators one can implement the easiest, using the basic components available)... A very interesting research project, and probably will be done at some point, but will take a while to get into real-world products, methinks.