[C++][FlightRPC] Refactor remaining methods into transport-agnostic handlers

[asfimport]

ARROW-15282 only deals with the data plane methods, we should figure out a generic way to handle the rest of the methods

Reporter: David Li / @lidavidm

Related issues:

• [C++][FlightRPC] Support non-grpc data planes (relates to)
• [C++][FlightRPC] Refactor auth, middleware into the transport-agnostic layer (relates to)
• [C++][FlightRPC] Implement a UCX transport (is related to)

_{Note: This issue was originally created as ARROW-15836. Please see the migration documentation for further details.}

[asfimport]

Todd Farmer / @toddfarmer:
This issue was last updated over 90 days ago, which may be an indication it is no longer being actively worked. To better reflect the current state, the issue is being unassigned. Please feel free to re-take assignment of the issue if it is being actively worked, or if you plan to start that work soon.