Link-local protocols, such as CDP, don't actually use a VLAN, which is really an end-to-end layer-2 construct. The frames that CDP uses are untagged so some people consider them in the native VLAN, but they can never go beyond the next hop because they are confined to the link.
If a frame is untagged, and can't be propagated beyond the next hop, it is really in a VLAN? No, because if it was in the native VLAN it could be propagated to the other ports in that same VLAN, and that is not the case for the link-local protocols. The fact is that you could have no native VLAN set for a trunk and remove VLAN 1 from the list of allowed VLANs on the trunk, and CDP,etc. will still cross to the next hop.
We do what you want to do all the time. You can't actually remove the layer-2 VLAN 1, but not allowing VLAN 1 across the trunk essentially disables it. You can restrict the trunk from using VLAN 1, and don't define a native VLAN for the trunk. You can remove the layer-3 SVI for VLAN 1, and add an SVI for whichever VLAN you wish to use to manage the switch. The SVI can have the IP address of the switch that can be used for switch management.
VLAN 1, as the default VLAN, is now considered a security risk, as are any native VLANS. After removing VLAN 1, and any native VLAN, from the trunk, you have crippled the ability of an attacker from using a default that is present in most networks. Untagged traffic will not be propagated beyond the link.