I believe that is how on site notifications work.
They’ll sit till the stream is confirmed to be up, which will align with when the webhook for stream up occurs. Which will be anything from 30 seconds to 3-5 minutes after the video starts.
As to not getting it, if Twitch thinks you are already on that streamers page then it won’t tell you about that stream since you are already on the page I believe.
Asking about first party stuff won’t get much of a answer on this forum since we are all third party round here. shrug