The previous fixes do not go far enough. I've updated in a similar fashion to Riften. Unfortunately, unlike Riften, there's no script with numbered conditions, so the 24 hour timeout will have to suffice. (More like Riverwood or Whiterun.)
The randoms need to be conditioned at 24 hours, as are other innkeeper rumors. Technically, you don't need that for four starting quests, but did those anyway for consistency. Likely slightly faster condition checks, although probably not enough to matter.
"Haven't heard anything lately, but check back with me... you never know." is mistakenly included in the random section, so it will play even when there are plenty more rumors. The RE needs to be on the previous record (previously fixed by #26700), so that it only plays after there are not any random rumors. It needs no timeout.