Binance and other exchanges also have P2P marketplaces -- I believe you don't need KYC to view these. Many leave their contact details, least last I checked.
It depends on the quantity of crypto currency the OP wants to trade. Binance in particular would require KYC for some certain amount of crypto currency traded in their platform and if the OP is to trade a large quantity then he will definitely need to do KYC to increase his limits of withdrawals and deposits. But if you read his statement carefully you will see that the OP wants a non custodial wallet that doesn't require him to be Kyced or uploaded any of his identity or any other details. What he needs is a decentralized exchange and I believe some persons might have dropped some for him, so it is left for him to research on them and choose the one that suits his interest.
OP sounded like he needed more than just the direction to get BTC , but perhaps a trusted person who would volunteer to help him out and also because as at now, it would be hard to acquire BTC without KYC verification even the wallets, one needs to be equipped to meet the limiting challenge that comes in form of fees for withdrawal and transfer and transactions limit.
Unless there's the plan to buy small, then the sites where to get BTC and get non custodial wallets has been given here as I have read.