Everything below is what I have learnt from others so please take it with a grain of salt and correct me if I’m wrong on anything.
Cons: riskier, more work, individual temps/humidity.
Pros: cleaner, easier to work with, one set of temps/humidity.
In the wild many baby’s won’t survive even pipping out of the egg due to many environmental and physical factors. If a mother stresses out for any reason then the eggs are at risk of rolling/moving/being crushed.
If you take the eggs and replicate the exact environment they need then you have removed the risk of damage and also the risk of being bit whenever you check up on them.
While the mother has the eggs with her she will stay in protection mode
Also if the mother poop/wees it not going on the eggs and improves the overall sanitation of the process.
Something I don’t hear said much is that when the clutch has laid, you have mere hours to separate the eggs otherwise they will fuse together and you can risk tearing the shell afterwards. If you miss this window then they should be left until cutting/pipping day to separate.