How can I ssh into a remote machine so that the remote machine shares the client’s internet?

I have the following setup:

┌────────────────────────────────────────────────────────┐
│                        Internet                        │
│                                                        │
│                 ┌────────────────────┐                 │
│                 │                    │                 │
│                 │                    │                 │
│                 │       client       │                 │
│                 │                    │                 │
│                 │                    │                 │
│                 └─────────┬──────────┘                 │
│                           │                            │
│         ┌─────────────────┼──────────────────┐         │
│         │                 │                  │         │
│         │       ┌─────────┴──────────┐       │         │
│         │       │                    │       │         │
│         │       │      public        │       │         │
│         │       │       server       │       │         │
│         │       │ pub.ip:123.11.22.33│       │         │
│         │       │ pri.ip:10.0.0.2    │       │         │
│         │       └─────────┬──────────┘       │         │
│         │                 │                  │         │
└─────────┼─────────────────┼──────────────────┼─────────┘
          │                 │                  │
          │       ┌─────────┴──────────┐       │
          │       │                    │       │
          │       │      private       │       │
          │       │       server       │       │
          │       │      10.0.0.3      │       │
          │       │                    │       │
          │       └────────────────────┘       │
          │                                    │
          │              Intranet              │
          │             10.0.0.0/24            │
          └────────────────────────────────────┘

The client ssh’es into the public server in order to perform maintenance on a private network. The private network is not connected to the internet except the public server as well as an additional public-facing http(s) proxy server with a separate public IP for web traffi (not shown), and the public server only listens on SSH, RDP, and VPN ports from the internet, as its only purpose is to serve as a landing server to access machines on the private network.

From the public server, I can further ssh into any machine on the private network. Currently, in order to update the machines on the private server, I have to open an internet connection for each machine, which is cumbersome. I’d like to share the internet connection of the public server with machines on the private network but only when I’m using the public server to ssh into the a machine on the private network, so that only when I’m performing working on the private machine does it have internet access. I would also like to do this without having to open and then close a tunnel via a separate connection, such that the internet connection is shared within the same session as the remote terminal.

Note that the ssh access is one-way, i.e. the machines on the private server cannot access the public server (or each other) via ssh (therefore also sshuttle) because the public server is configured only to accept ssh connection from the internet and not to accept connection from the private network.