Vlad If you're okay with having to manually convert the payments back to fiat, I'd recommend using a self-hosted solution like BTCPay because it's absolutely best for privacy. Remember that cryptocurrency is decentralized, so you don't actually need a middle-man to process your payments, although it might a bit be harder to setup than the hosted ones.
Now, I don't actually know that much about the different hosted cryptocurrency payment providers (so take this with a grain of salt), but it doesn't really seem like the community endorses Coinbase. Looking at Bitcoin, I did find a community-made list called Awesome Bitcoin payment processors which covers both hosted and self-hosted. Monero has on their own site a list of payment processors if you want to support that as well. I can't really say anything about the individual payment processors though because I don't have any experience with them - if anyone here is more knowledgeable, please come with your own recommendations.
If you're going with a hosted solution, I'd recommend just checking if they require the user to identify themselves, sign up, or create a wallet through their service before paying, because in that case I'd absolutely not recommend it. Otherwise, I think it's probably good, but check what the community has to say about it first (your search engine's "Forums" lens is great for this)
