Visualizing a board with 64 individual squares isn't necessary to see 20 moves ahead during a real game. I think what you're doing it's more practical, i.e. looking at a board and trying to visualize where all the pieces are... because that's what we do during a real game.
In Tisdall's book "Improve Your Chess Now" he suggests a technique where you visualize as many moves as you can, and as you start to reach your limit, burn that position into your mind (memorize it). Once you have, move on a few more moves from there. I've used this to visualize solutions to long tactical problems / studies (22 ply) and I think this would help you see 20 moves ahead. Probably not on the first day heh, but as you consistently practice it, you should get better.
In any case, I'm a believer in practicing the skill you want to improve.
One exercise for blindfold chess (but I think you can use it with your big board) is to imagine three pieces (like a queen, rook, and knight) on an empty board. Indicate which pieces protect eachother / which are undefended. Then move one of them (in your mind) and again say which pieces are protected. Repeat as long as you want. You could prepare a few of these on flashcards for yourself (the moves and the answers) and then attempt them.
I believe that being able to see a clear board in the head can really beneficial to a person's calculation ability. Problem is, I've been trying to do so for quite awhile and have had no luck.
I either: A) Can't keep all the squares in their place.
or B) Can't tell the distance between pieces.
I am trying a few things to help me at least be able to play twenty moves. I have drawn a large scale chess board on cardboard and attached it to my wall. There are no pieces on the board so I have to imagine them in my head. The board is simply there to provide me the guidance to tell the distance between the pieces and help me memorize what the board looks like so I can see it with my eyes closed.
I would greatly appreciate any links to videos, book suggestions, and/or maybe a method you used that has helped you. Or maybe you're here because you want to tell me that putting a board in your mind is useless...