Why do integer multiplication algorithms not use lookup tables?

It seems to me that we can use lookup tables for the multiplication of two large integers $ log (n) / $ 2, and that the number of entries for each table of these numbers should be $ O (n) $.

Now, multiplying two $ n $Naturals two-bit together can be assisted using a large FFT $ O (n / log (n)) $. It seems to me that we can then perform the multiplications and additions for the FFT using the look-up tables.

I wonder why this method does not have or seems not to attract attention. Is it because of memory bandwidth issues?

It seems to me that the use of tables would be faster, so I hope someone can provide me with better intuition.