Yes, the center speaker plays all sounds
that also occur in both
the L&R front channels. That's the sounds 'mixed to mono' by the recording engineer, the sounds that are perfectly 'centered' on the 'stage'. That includes all the SYNC tones and voices, which are actually in the L&R front, too (except the backup proximity alarms, if you have that, which are on the rear channels only).
If you disconnect that speaker you will not lose any
sounds, they'll be heard from the L&R channels and 'combined in your brain' to sound like center mono.
suggested simply unplugging the center channel and I found I liked that better, too, with my Alpine setup. I think that's because of the poor relative fidelity of the center speaker compared to the upgraded speakers. It's an easy thing to test, costs nothing and won't hurt anything.
You, too, may find that your preferred solution, especially if you mainly listen alone and use the 'driver only' occupancy setting. The center comes more into play when you use the 'driver+front passenger' occupancy setting, to help create a less-compromised 'stage ambience' for both folks in the front seat (a time-alignment function).
It's a bit harder to find an 8 Ohm speaker upgrade for the center that fits (the OEM center is 8 Ohm nominal unlike all the others). You can probably/maybe
run a 4 Ohm without damage/overheating of the output transistors over time, but it may be harder to 'balance' the center to the sides when you do that.
Yes, I'm back to a center channel with my different current setup, but that's not using the OEM center channel output. My new center channel is 'created' by my aftermarket DSP from the front L&R signals and I'm using a much more difficult/expensive 2-way active XO tweeter+midrange custom installed in the center so it isn't a fair comparison to the choices you have.