Proxying Nginx to S3

proxy-s3.nginx

 location ~* ^/(.*) {
  
    set $s3_bucket = "<bucket-name>.s3-website-eu-west-1.amazonaws.com";
  
    proxy_http_version     1.1;
    proxy_buffering        off;
    proxy_ignore_headers   "Set-Cookie";
    proxy_hide_header      x-amz-id-2;
    proxy_hide_header      x-amz-request-id;
    proxy_hide_header      x-amz-meta-s3cmd-attrs;
    proxy_hide_header      Set-Cookie;
    proxy_set_header       Host $s3_bucket;
    proxy_set_header       Authorization "";
    proxy_intercept_errors on;
    proxy_pass http://$s3_bucket/$1;
    break;
  }

shouldn't it be set $s3_bucket "<bucket-name>.s3-website-eu-west-1.amazonaws.com"; (no =)?

Yep, that will be more syntactically correct. (Full disclosure, this file is 4 years old and I haven't touched this since).