How much would it help a GM to be able to move pieces while thinking?

If a GM could run through variations on a board (or more likely computer screen without access to an engine), how much would his/her rating actually improve approximately?  We all know that GMs have tremendous abilities to visualize.   One way to think about this is how much did it help a GM to be able to analyze a game in his hotel room at night when a game was temporarily suspended in the days before chess engines became dominant?  Of course, in those situations, in addition to being able to move the pieces, they also had much more time to think, access to their seconds, etc.

I expect, given the audience I am asking this question to, that almost everyone will say that there would be minimal advantage.  But still curious if there are contrarians out there who think this would be a big advantage.

Quite a lot, I would think. Ultimately, every loss is due to getting a position you weren't expecting to get. A strong GM would know perfectly well what variations he/she should look into, and the probability of getting strongly unexpected positions would go down significantly.

I would expect it to definitely help.  You see many old pictures and some videos of top players using mini/travel chess sets to study positions. Now how much it would help is the question. 

But the ultimate result of this incorrect evaluation is getting a position the player didn't expect to get. Whenever one makes planning and considers different moves, they envision the approximate consequences of those moves. A strong player makes a move with a favorable eventual position in mind, and when they get a completely different and non-favorable position, it is because they didn't know their move would lead there.

Now, if they were able to actually move their pieces around while thinking, then, once deciding on a move, they could play out a few variations and see move flaws with their ideas. 

It is like daily games versus live games. In daily games, until it is one of those days when I'm not feeling like spending much time on chess and end up rushing my moves, I always, before making a move, play it out in the analysis mode and try to envision the possible continuations. Very rarely do I get something completely unexpected, mostly the mistakes I make is in the nuances that become essentially much later on - but, at the very least, some basic plan elements usually match what actually ends up happening, because I've already played it out before and tried to anticipate all the surprised.

Grandmasters would have the extra advantage, since, unlike me, they are much better at selecting proper candidate moves: something that I have to play around with for a while to see that it doesn't work, they will see right away. Their analysis will be much deeper, and it is very-very unlikely that they will miss any tactics or make a positional mistake that will turn out to be crucial a few moves later. Most likely, the mistakes they will make will only manifest 8-10 moves later, making their overall play much more sturdy.

While I do think it would help GMs, I can assure with 100% certainty it would help this patzer.

Wouldn't it be funny if an OTB tournament made an exception and all the players would have at their discretion a small chess set to next to them to play as an "analysis board" while the game was being played?  

In this tournament all players would have their official board with the clocks running, and right next to them they would have their little chess sets that they would fiddle around with before going to the official board, making their move, pressing down the clock, and writing down the moves.

Just do it once.  See what it would be like.

I know for damned sure it helps me, a lot, when I'm playing a casual, yet relatively serious game against an engine. I'm at home, and I am training, so it's okay, but I understand it doesn't really replicate tournament conditions because it does make a difference. You might be a pretty sharp calculator and good visualizer, but nothing like seeing the board in front of you. Even GMs are human, and it would probably give them the equivalent of an addition 75 ELO points of strength if they could freeze time and do this.