Classical music (particularly the baroque and classical periods) for sure grooves, and everything still needs to sync up to the same grid, but it’s definitely different than in jazz, folk, or pop music since the tempi can be much more fluid.
I think what can trip people up who haven’t played in an orchestra much but maybe do have experience in Broadway pits is due to the sound delay between the different parts of the orchestra: the back of the orchestra (brass, but also sometimes percussion) has to pretty much be right with or even a little ahead of the conductor because if they play at the exact same second as the first violins, to the audience they will sound late. Likewise, the strings will often play a little behind the conductor to compensate for the time delay. It’s just something you intuitively learn as you play in an orchestra more frequently. In a Broadway pit, it’s usually a smaller ensemble in a smaller physical space and everyone is mic’d or plugged in and running through the PA, so the sound the audience hears is largely not what’s happening on or below the stage, but what’s being pumped into the house. In that situation, everyone (the instrumentalists, conductor, and actors) needs to be on top of the beat because there’s basically no sound delay to deal with.
That being said, I do think in non-baroque contexts classical musicians are generally not taught how to “feel” music in the same way as jazz or pop musicians. I wouldn’t necessarily classify it as more reactive because if you see a chamber group play you can definitely see that they’re all very much feeling the same collective pulse, but it’s not the same as playing with a drummer and/or bassist and trying to line everything up with what they’re doing. What classical musicians do in small groups might be more cerebral than what groove musicians do, but I’m kinda just spitballing.
But also classical musicians are just not taught how to swing and when it comes up for us if our teacher isn’t familiar with jazz they’ll tell us to just play a 1/4 note triplet and an 1/8 note triplet without necessarily getting into how accents also create a swing feel. It also leaves out that a swing feel can range from almost straight 1/8 notes to a dotted 1/8 + 1/16 depending on the context and that thinking about it in strict metric terms kinda defeats the point.
I feel like I’m rambling, but I hope this can offer some enlightenment to people reading this thread.