Wood is going to move. It is going to swell and contract with humidity changes in the home. This typically happens during the winter and summer, when the temperature changes outside cause the heat in the home to dry out the air and during the summer the air in the home typically, is more humid. Wood WILL react to this, there is no stopping it.
In my opinion the floor is narrow enough not to warrant gluing down to the OSB, secondly on that, OSB is not the best material to be gluing strip flooring to. Depending how the OSB weathered as subfloor while it was exposed to the elements, the top layer of chips/flakes are barely set in place still, not to rely on for gluing. Beside, you don't need it, But if it is used, it won't cause any problems, it just won't do anything either.
The biggest factor in laying hardwood floors is the humidity that is present in all parts of the process. This is where a moisture meter comes in. Any flooring installer worth his salt has a good one. IMHO, if he doesn't along with what he is prescribing, he would not do my floor.
Since the wood is going to move, and the T&G joints are glued, the flooring will crack elsewhere. Instead of the joints opening and closing the minute amount, when the floor is laid in the proper conditions (this is where the moisture meter comes in) which would be undetectable, with gluing, the wood would crack in irregular places, creating a mess to see on the floor. If the conditions are not right for a floor, I have seen a white oak floor shrink to an 1/8" gap over much of a floor laid where the wood was not acclimated to the house first and laid without consideration for its' moisture content.