The conductor arranges the piece. There’s a LOT of interpretation to written music. A time signature indicates the structure of a measure. How fast it’s actually played (overall) is often indicated, but a conductor may choose to alter the tempo.

Then within the sheets there are indicators for louder/softer, faster/slower, and a number of other elements for each instrument and the music overall, but again the conductor will choose what that means for a given piece, at a given stage, at a given time (maybe there are more people in the audience, so he increases volume with hand gestures, or it’s a warmer night at an outdoor venue, which means less dense air so the sound carries less).

The reason the conductor bows is the sum total of the music is a result of his choices, his direction of the musicians. The resultant music as a whole is because of him. No small task.

He also represents the orchestra as a unit. That bow represents every musician there.

When they excavated Beethoven’s tomb, he was randomly crossing out notes on a sheet. Badly decomposing.

Among other things, the conductor “counts time” to make sure everyone keeps the same beat. Like a human metronome in front of the band.

The conductor stands in front of the musician, so they can hear the integrated sound of the band/orchestra. When you are sitting in the drum section, or with the trumpets, etc — you cant really hear anything but yourself. The conductor has to act as the ears of the entire thing to make sure the sounds are integrating properly. Much like the monitors a rock band uses. This is the pointing, lifting, and dampening motions you see integrated into movements made with the baton.

What you aren’t seeing when you are watching him shake his baton is all the prep that went into this before the actual performance. The conductor’s movements are “reminding” the performers what they worked on/discussed during rehearsals. Much like the director of a movie directs the actors, cinemetographer, etc. This is him reminding a performer or section to play louder, softer, whatever, for that section of the music like they talked about earlier.

So when you see the audience applauding the conductor, they are applauding all the work that went into it. both during rehearsal and at the final performance.

Hope this helps