Can someone choose to keep their email visible in whois if they are a private individual?

(Also, while registering a domain lately, I didn't see an option to uncheck this feature.)

Keeping your email visible in WHOIS is typically not something individual users should want to do. When you register a domain name, your information goes into the WHOIS database, which is publicly accessible. This means that anyone can look up the details of domain registration, including the name, address, and, in some cases, email of the registrant.

Many registrars offer something called WHOIS privacy protection, which masks your personal information from being displayed in public records. If you don't see an option to uncheck this feature when registering a domain, it might be because the registrars have it enabled by default for privacy reasons. Some registrars may also charge an additional fee for this service, which can be a reason why it wasn't obvious during the registration process.

For private individuals, exposing your email in WHOIS can lead to spam and unwanted contact, which is why most people opt for the privacy protection. However, if someone really wants to keep their email visible and doesn't see an option, they could try to contact the registrar directly for assistance. In rare cases, there might be specific domains or country codes that do not allow for privacy protections, which could be the reason behind not having that checkbox.
While it's technically possible for an individual to retain a visible email in WHOIS, it's generally not advisable due to spam risks and privacy concerns. Always check your registrar's options and policies related to privacy when you're registering a new domain to ensure you're making the best choice for your online presence.

From what I understand, payment for Privacy Protections is not included within the domin fee. Also, it's really important to have this feature for keeping your personal information safe and secore online.

When it come to com-net domain zones, the registrar is the one who keeps all the user information, not the registry itself. Also, the whois information is presented by the registrar too, which is important for privacy concerns.
So, if you're planning to register a domain, be sure to check out how your chosen registrar handles your data.