Why don't integer multiplication algorithms use lookup tables?
If you want to use lookup tables, and you have 4GB of memory, you'll only be able to use a lookup table with about $2^{32}$ entries or fewer, so you'll only be able to handle multiplication of numbers that are at most 16 bits long.  If you want to multiply larger numbers, you won't be able to use lookup tables for multiplication.  Typically we want to multiply numbers that are larger than that.  And when we do want to multiply small numbers, using the CPU's built-in multiply instruction is usually faster than using a lookup table.

So there is no size (that I'm aware of) where a lookup table is a good solution for multiplying two numbers.

Now you're thinking that maybe the numbers that are generated by FFT should be multiplied with lookup tables.  But those are just numbers, and as I said above, there is pretty much no situation where the fastest way to multiply two numbers is with a lookup table.  It doesn't matter how those numbers were generated -- whether by FFT or by any other process.

So I'd say the reason it doesn't have more attention is because it doesn't deserve attention; it seems unlikely to be faster or better than existing schemes.



Lookup tables are occasionally used in cryptographic code, where constant-time access is needed.  This is not because they are faster -- they're not; they're slower -- but because they have other properties that are beneficial.  I don't know of any specific examples, but it wouldn't surprise me to discover that someone had used a lookup table for multiplication in some constant-time cryptographic code.