FPC broken for logged-in users as of Magento 2.4.8

[rhoerr]

Preconditions and environment

No response

Steps to reproduce

Expected result

Actual result

Additional information

Port fix from https://github.com/magento/magento2/issues/40039

Release note

No response

Triage and priority

• [ ] Severity: S0 - Affects critical data or functionality and leaves users without workaround.
• [ ] Severity: S1 - Affects critical data or functionality and forces users to employ a workaround.
• [ ] Severity: S2 - Affects non-critical data or functionality and forces users to employ a workaround.
• [ ] Severity: S3 - Affects non-critical data or functionality and does not force users to employ a workaround.
• [ ] Severity: S4 - Affects aesthetics, professional look and feel, “quality” or “usability”.