We have no pictures to see what the garage looks like but if the ceiling has exposed beams that could be a real feature and it will add a sense of height. Nothing else to go on without a photo.

It also depends on how you intend to use the room.

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Basement ceilings IMO have too much wiring, plumbing to look attractive exposed why not just drywall it is only 1/2 “ thick maybe the contractor is just lazy.

If noise is a consideration, then cover it. To be sure, go down there, sit in a chair, turn the lights off and "listen". Get the teens to do their standard roughhousing with game play and some rowdy yelling and see what happens. If the ceiling is covered right now, imagine it being close to twice as loud if it is exposed.

The other thing about exposed ceilings: dust and debris often fall from the subfloor. This would have been caught by the drywall...but with exposed subfloors you get the chance to see 3 year old dust bunnies dropping into your lap. Takes a pretty iron stomach to handle that idea.

Of course there are ways to "leave the look" yet completely enclose the space...it is pretty but expensive. It takes sistering the beams and cutting/fitting plywood/OSB up into the spaces between the joists. It gives the look of exposed beams without being exposed. The labour and added material can double the cost of a traditional ceiling....but it does allow you to add acoustic insulation without seeing that insulation. And it stops the dust bunnies from dropping in for lunch.