Two nasty surprises in Home Assistant's config
blog.frankel.ch
blog.frankel.ch
Firstly, issue #153175 refers to the "time of day" sensor, which is not what you're using - you're using the "time" trigger. The TOD code does a bunch of timezone acrobatics that can be fixed, but I don't think it'll affect your problem.
Secondly, I built a tiny regression test on dev (on top of 333a3421) to reproduce your problem, and it didn't repro. I created a "trigger: time, at: 06:00:00, Europe/Paris", attached it before the DST transitions, and checked when it fires before, during, after the DST actual. The trigger fires perfectly on 06:00 walltime, and adjusts utc to match.
Thirdly, the HASS UI has a small bug where it shows "Paris/+1" even if it's currently DST. It's a UI bug only, but it can lead to someone thinking that the entire HASS core doesn't respect DST.
Silly tech support questions - are you on latest? Did you try playing around with {{now()}} in developer templates? Maybe open a new issue?
IMHO stalebots are an anti-feature in an issue tracker, and do more harm than good, but that's a different discussion.
Maintainers ignored the issue for 3 months until their own automation closed it. It wasn't a rogue github bot.
EDIT: I'm not finding the issue that I'm thinking of very quickly. But it was a bug report from 2022 which they looked at at the time, said "yeah, this is an issue we'd like to fix but we don't have a good fix right now", and then the issue was silent. Later on, Svelte went through a major internal rewrite between Svelte 4 and Svelte 5. With Svelte 5's new internals, they were able to fix that 2022 issue. I noticed it because I had just recently read the Svelte changelog, idly clicked on some of the PRs, and noticed an issue with a very low number that was marked as fixed. But I'm not finding it now.
Issues waiting for far longer then 3 months before they are fixed is super normal of important but not a blocker category of bugs.