reverse-proxy with nginx: using an existing https site as backend

I have one site somewhere that's accessible on the internet as "https://site.example1.com".

I also have one another site accessible on the internet as "https://www.example2.com".

I have full control over both sites and they both have valid Certificates.

Now I want to have the first site to be accessible as a location on the second; this means that "https://www.example2.com/site" should display the contents of "https://site.example1.com".

I tried a plain:

server {
    listen 443 ssl;

    resolver 127.0.0.1 8.8.8.8;

    server_name updates.okcash.com;
    ssl_certificate /etc/letsencrypt/live/www.example2.com/fullchain.pem;
    ssl_certificate_key /etc/letsencrypt/live/www.example2.com/privkey.pem;

    ssl_verify_client optional_no_ca;

    location /site {
        proxy_pass              "https://site.example1.com";
        rewrite                 /site/(.*) /$1     break;
        proxy_redirect          off;
        proxy_set_header        Host               $host;
    }
    location / {
        proxy_pass              "http://localhost:5001";

        proxy_connect_timeout   60;
        proxy_read_timeout      86400; # 24h
        proxy_send_timeout      60;
        proxy_intercept_errors  off;
        proxy_http_version      1.1;
        proxy_set_header        Host               $http_host;
        proxy_set_header        X-Real-IP          $remote_addr;
        proxy_set_header        X-Forwarded-For    $proxy_add_x_forwarded_for;
        proxy_set_header        X-Forwarded-Proto  $scheme;
        proxy_set_header        Upgrade            $http_upgrade;
        proxy_set_header        Connection         "upgrade";
    }
}

but this does not work well because browser (accessing "https://www.example2.com/site") is presented server certificates from backend ("https://site.example1.com") which are invalid for the request.

How can I force nginx front-end to send out its own certificates and verify backend identity on its own?