SNOW-1418523: Remove session variable that can be local

[sfc-gh-aalam]

1. Which Jira issue is this PR addressing? Make sure that there is an accompanying issue to your PR.

   Fixes SNOW-1418523

2. Fill out the following pre-review checklist:

   • I am adding a new automated test(s) to verify correctness of my new code
     • If this test skips Local Testing mode, I'm requesting review from @snowflakedb/local-testing
   • I am adding new logging messages
   • I am adding a new telemetry message
   • I am adding new credentials
   • I am adding a new dependency
   • If this is a new feature/behavior, I'm adding the Local Testing parity changes.

3. Please describe how your code solves the related issue.

   Minor refactor to handle thread-safe session. We remove _tmpdir_handler which is only used inside method _upload_unsupported_packages.