Both the bundled email provided by your domain registrar and an "external" email provider like Gmail can be safe and secure, provided that appropriate measures are taken. The decision may depend more on the specific features and conveniences you are looking for, and your trust in the company providing the service. Here are some points to consider:
1. **Security Features**: Google, as one of the largest tech companies globally, offers robust security features such as two-factor authentication (2FA), machine learning spam filters, and strong data encryption. Not all domain registrars might have the same level of advanced security measures. You should investigate whether your registrar's email service provides similar protections.
2. **Reputation**: You mention that your registrar is a multinational with a good reputation. That's a positive sign, as established, reputable companies typically have good security practices in place to protect their customers. But do remember, bigger doesn't always mean safer. You'll still want to investigate the registrar's specific security features and practices.
3. **Data Protection and Privacy Policies**: It's also important to review your domain registrar's and Google's data protection and privacy policies. Different companies might have different policies about how they handle and protect your data.
4. **Convenience and Usability**: Google's Gmail is known for its user-friendly interface, powerful search features, and easy integration with other Google services (like Google Drive, Google Docs, Google Calendar). The bundled email service from your domain registrar might not offer the same level of integration or ease of use, which could be an important factor depending on your needs.
5. **Professionalism**: An email address associated with your own domain name can appear more professional than a generic Gmail address, especially if you're using it for business purposes.
6. **Control**: With your own domain-based email, you have more control. You can easily switch email hosting providers without changing your email address. With Gmail, you're tied to Google's service.
As for your question about whether an email address bundled with a domain could be more easily exploited by hackers, it really depends on the security practices of the domain registrar. The same risks exist for any email account if not properly secured, such as being targeted for phishing attempts or other forms of cyber-attack.
In conclusion, there's no one-size-fits-all answer. You'll need to evaluate your specific needs and the offerings of your domain registrar's email service and Google. In terms of security, either can be safe as long as they have strong security measures in place, and you follow best practices like using strong, unique passwords and enabling two-factor authentication.