> “The policy is applicable to all County issued devices, including desktops, laptops, servers, tablets, cellular phones, or any other information processing asset— including any and all devices connected to the County of Lancaster network for the purpose of accessing County of Lancaster data or services.”

You're reading that too broadly. So long as the device itself doesn't connect to the County's network, it's not subject to this rule. So yes, you can still check email on your personal device and not worry about running afoul of this policy, so long as it's not connected to the network.