Yes, bitcoin core fits well with what I have described, but I would like the user to have the opportunity to use other currencies(4-5 popular currency ada, eth, usdc), unfortunately, the specifics of the my site do not allow using the payments services(coinpayments or btcpayserver) that you mentioned. at the moment I'm leaning towards binance with its ability to create sub-accounts in a business account via api.
Are you developing your own website or are you using CMS? I don't know what do you mean your site does not allow coinpayments or BTCpayserver but I already have experience using them with WordPress. You can achieve what exactly you want in BTCpayserver someone I heard that they could make their customer have a single address each customer through WordPress and I'm sure you can also do that if you are a web developer or just read the full document on how to implement it. I'm not sure in Coinpayments their payment solution is only for receiving that support multiple crypto.
About using ETH, ADA and USDC I think you can't able to add it on BTCpayserver I never heard someone succeed adding these Coins except on other coins that are similar to BTC.
Anyway there is a list of payment processor you can try to check them and maybe one of the payment processors support your website.
Here's the link
https://github.com/alexk111/awesome-bitcoin-payment-processors