Ok, so I'll attempt to clarify
BTCpay is designed as a drop-in replacement for Bitpay, that way merchants can ditch Bitpay with minimal difficulty (hence the API compatibility).
But BTCpay isn't a payments processor, just a DIY software interface. The idea is that the merchant gets an account at a BTC exchange, then uses that account to sell BTC from their customers on whatever exchange they like (automated like Bitpay, the customer sends BTC direct to the merchant's exchange account when paying for goods). This has the advantage of getting around Bitpay's cash-out limits, protecting customer privacy and minimising risks (no reason why more than 1 exchange couldn't be used).
BTCpay will likely integrate Lightning payments fairly quickly too, at least 1 BTCpay developer is a frequent contributor to one of the Lightning implementations. How soon Lightning happens with Bitpay is anyone's guess, they have apparently been trying to make life more difficult for their customer base of late. It wouldn't surprise me if Cheapair's action here is a direct consequence of Bitpay's bizarre behaviour (Cheapair made their name accepting Bitcoin, so presumably alot of their customer base depended on being able to use Bitpay, which is no longer easy).