What is the most reputable Bitcoin payment processor that has privacy friendly terms and conditions? BitPay and Coinbase are known to have used chain analysis to spy on customers. What are the alternatives for reputable merchants who do not approve of this type of privacy violation applied to their customers?
I'm pretty sure that all crypto payment processors that deal with fiat <-> cryptocurrency exchanges are doing AML/KYC routines including chain analysis. That's not only true for BitPay and Coinbase but Circle et al. too. As soon as you're trying to move value between crypto and fiat you're going to have to deal with different levels of privacy invasions if you're using a payment processor. The exceptions to those KYC/AML routines is either cash <-> crypto transactions (like localbitcoin, mycelium...) or using a crypto currency that can't be traced by (evil) companies like chainalysis.
tl;dr I don't know any non-privacy invading bitcoin payment processor as soon as fiat gets involved.
ps: However I know that there a various people working on new (open-source) cryptocurrency payment processors. I will try to dig out some additional info on those and update my post accordingly.
edit: interesting update on coinbase: https://news.bitcoin.com/coinbase-helps-fbi-shut-kickasstorrents/
You need solutions without third parties or decentral solutions for privacy because central/third parties are regulated or might be corrupt with your data. A good payment processor without third party involvement is btcpayserver.org . It manages BTC (plus lightning plus litecoin) payments fully automatized. In case you want to exchange it into fiat without a central party having knowledge about I'd recommend bisq.network . The bank can see of course the fiat transfer but has no clue about the payment reason. So far this 2nd step isn't working automatically, however the DAO (decentralized autonomous organisation) bisq is working on an API. bisq transactions in the blockchain are difficult to trace because they are like payjoin transactions.
Privacy-wise the only way to go is to self-host payment processing. Fortunately, as of 2021 good end-to-end open source solutions are available:
The worse alternative but still no KYC are the so called "watch-only" payment processors. The watch only payment processor is privy to all transactions in your wallet. The two I am aware of (never tried though):