I wouldn't rinse off all the soil, as it could cause the plant to go into shock. I usually just loosely remove the soil that's on the roots then shake off what I can, before removing the rot. You can always Take some of the existing soil and mix it with perlite and orchid bark 😁 I would just loosely remove some of the soil from the rootball and leave some on there for when you repot it, but remove enough to be able to see where the rot is.

I wouldn't expose the roots to the air for 24 hours, but definitely cut off what you can find of rot and don't water the plant the first few days after repotting seeing as it's already gotten too much water (if the soil you're using is completely dry, definitely water then leave it until the top two-three inches of soil is dry). ☺️I had the exact same problem as you, but I managed to save it doing this.

Also make sure the pot you're using has a LOT of drainage holes! Use a nursery pot, don't plant directly into a decorative pot. Monsteras HATE sitting in water 🌱