Jira (both Cloud and Server) does cache permissions, heavily. Jira effectively has a local cache of user and group relationships for all users who have access to a given blah.atlassian.net Jira, and caches lookup results in a shared cache too. It's complicated by having well over a hundred thousand active Jira sites spread over several hundred stateless servers, which does make effective caching a bit trickier.
AFAICT Jira Cloud's slow feel currently stems largely from the front-end, amplified by the debacle that is modern front-end engineering. It's tricky to get 300+ engineers to ship fixes and enhancements in a multi-million line codebase over 15 years old on a good day. Feature creep in design systems, constant ecosystem churn, and poor build tools all conspire to make it that much harder to prevent performance regressing.
But even knowing all that - yeah, I wish that Jira's performance was 5x better too :)
Disclaimer: am still employed by Atlassian but all views are my own, things may have changed a bit since I haven't touched any Jira code in ages, etc.