Would you just stick to the first three exchanges due to this?
Yes. I stopped using LocalBitcoins long ago precisely because they turned themselves in to a centralized exchange demanding KYC from all their users. If privacy is what you are looking for, then you must avoid LocalBitcoins.
Bear in mind that some traders on P2P exchanges may still request personal details themselves, even if the platform does not. Sometimes this is necessary, for example if you are trying to sell bitcoin in return for a bank transfer or a money order, as the they need to know where to transfer the fiat or the money order must have your details on it for you to be able to receive it. It should only be the bare minimum amount of information in this case. Sometimes, however, traders will request your details and even copies of your ID as part of their security practices to try to reduce the risk of theft or scamming. Up to you whether you want to trade with such users, but I would suggest that giving random individuals your documents is even riskier than completing KYC at a major exchange. These users should be upfront about their requirements and therefore easy to avoid.
