rpc: Update server to use new RPCHelpMan

Summary:
Also, move Check to inside HandleRequest

This is a backport of core#19386
https://github.com/bitcoin/bitcoin/pull/19386/commits/fa7592bfa8691eb0289b21da3571709a18391b0f

Depends on D10009

Test Plan: ninja all check-all

Reviewers: #bitcoin_abc, majcosta, Fabien

Reviewed By: #bitcoin_abc, majcosta, Fabien

Subscribers: Fabien, majcosta

Differential Revision: https://reviews.bitcoinabc.org/D10010