Tests relying on `reqwest::Client` may fail with "dispatch task is gone: runtime dropped the dispatch task"

Verification

• I searched for recent similar issues at https://github.com/rust-lang/rustup/issues?q=is%3Aissue+is%3Aopen%2Cclosed and found no duplicates.
• I am on the latest version of Rustup according to https://github.com/rust-lang/rustup/tags and am still able to reproduce my issue.

Problem

This is causing irregular failures for different CI platforms:

• Linux
  • https://github.com/rust-lang/rustup/actions/runs/9303236385/job/25605096671
  • https://github.com/rust-lang/rustup/actions/runs/9354772182/job/25748366479
  • https://github.com/rust-lang/rustup/actions/runs/9337787790/job/25700115067
  • https://github.com/rust-lang/rustup/actions/runs/9343224087/job/25712479916
• Windows
  • https://github.com/rust-lang/rustup/actions/runs/9335696949/job/25695378538

The error message here is identical (although the exact test that failed might differ):

running 395 tests
...
test suite::cli_exact::check_updates_with_update ... FAILED
error: could not download file from 'https://static.rust-lang.org/rustup/release-stable.toml' to '/tmp/rustup-updateD58GcC/release-stable.toml': failed to make network request: error sending request for url (https://static.rust-lang.org/rustup/release-stable.toml): client error (SendRequest): dispatch task is gone: runtime dropped the dispatch task

Notes

Possibly related to https://github.com/hyperium/hyper/issues/2312, https://github.com/Azure/azure-sdk-for-rust/pull/1550 or https://github.com/fussybeaver/bollard/issues/387#issuecomment-2008133973.

I'm pretty sure that #3367 has caused this.

Start with the suite::cli_exact::check_updates_with_update test and the reqwest::Client usage, then review the change introduced by issue #3367 and the linked CI failures. Reproduce the intermittent error on the affected platforms; done means the relevant tests no longer fail with a dropped dispatch task.