My carpet fitter says that synthetic is better than wool for cleaning in that you can use chemicals – they normally say that you can bleach them. Never had occasion to do this. Wool is generally better for wear as it doesn’t flatten as easily.
The feel and sawdust thing is probably more dependant on the type of carpet. I can’t remember the technical stuff but. We have some synthetic that is single strands and that cleans easily. The pile is relatively short but then you get looped types which things stick to like velcro.
FWIW we got a synthetic in our lounge. It was relatively expensive, the cheap ones were nasty. It has been fine to clean the odd mark off but we only used water so I assume wool would have been fine too. It has flattened slightly (in 18months) by the door. We are about to get the stairs done and we’ll be using wool.
If the carpet is for the babies room then just get a cheaper one and a nice rug.