There have been attempts at non-woke payment processors. But they've been shut down because banks and credit card companies refuse to do business with them.
You can't just throw up a payment processor website. You need the support of VISA, MasterCard, American Express, Chase, Citibank, Wells Fargo, etc.
If all the Big Finance companies boycott you because the ADL or SPLC accuses you of funding "hate," then your website won't be able to process payments.
Perfect use case for crypto...direct private person-to-person payments w/o extra 3rd party fees. To keep it private get a fresh crypto wallet (via VPN if really paranoid)...do not reuse a wallet address and do not use traditional “KYC” exchanges like Coinbase. (IOW do not tie your crypto addresses to any personal information).