OPSEC and user experience are completely unrelated things.

If you have something to hide (from anyone at all), you have to employ opsec measures. Using an email account that can't be immediately linked back to you is the most basic of them (and would be perfectly sufficient in the scenario described).

It wouldn't even matter if the website didn't leak in any way that an email is registered with them, because data breaches happen, and should one happen, you'd be fucked from that perspective anyway. Remember the Ashley Madison leak?