It can be mandated by application of 2018 GDPR. If a Cllr chooses to use their private E mail address, they must provide assurance to the council that they are complying with the act with respect to the way they process personal data and protect it etc. For many they may not be able to give that assurance due to lack of understanding about the act, and inability to apply control measures or set the security of the devices they use.
The ability of the clerk to read Council E mails will depend on the set up of their mail servers. If they are located remotely and managed by an external service provider, the clerk would have to stipulate remote access and permisions being given to them. If it came as part of a package with the Internet Service provider, the Clerk may be able to nominate themself as an administrator to gain access. If it was provided by the Principal Authority, that would be the most likely to be secure, and least open to unrequested access.