Question

After adding nameserver at Domain management > Nameserver management, they do not resolve across the internet. What is the purpose of this utility?

Answer

Nameserver management page exist for a very specific reason. One should take heed of the disclaimer at the top of the page and not try to create glue records for the domains that require such records to be created:

The 'nameserver' section allows you to create and manage nameserver objects. Use this functionality only if normal glue records in the domain registration and management sections do not suffice. Nameservers can only be created for domains that are registered in your Openprovider account.

Nameserver management page has a very distinct purpose - to provide functionality for registering nameserver objects at a given registry. Creating nameservers that way will not make them accessible in the DNS.

Let's consider the following scenario: the reseller has a domain portfolio shared across Openprovider and another registrar for any applicable reason (e.g. scheduled transfer sessions). The reseller has a domain example.tld which is registered within Openprovider.

In the possession of the reseller there is also a nameserver which is to be used for his/her other domains which are registered with a third-party registrar, and there's a need to name this server -  ns.example.tld. If the reseller was to assign this nameserver to a domain external-example.tld which is hosted within another registrar, he/she would receive an error akin to this one:

Nameserver creation or update failed:
ns.example.tld, 2201, Authorization error

In-depth explanation of that behaviour is available in the following article: Nameserver creation or update failed: 2201, Authorization error.

Naturally, the question arises: how does one assign ns.example.tld (Openprovider) as a nameserver to external-example.tld (non-Openprovider) without violating the registry's object permissions policy?

Here's where the Nameserver management functionality comes into play. It allows creating a nameserver host object in the registry (i.e. it will literally register a nameserver there) in order to grant the third-party registrar an option to assign a nameserver(whose name is registered within OP) to a domain which is registered within that third-party registrar. 

Hence, mere nameserver object registration is the intended application of this utility, rather than actual DNS resolution.