H2 is a baseline. If players behave consistently acceptably by their peers, it's no effort at all to regain H5 status. H5 is also not intended to be a permanent status, as it's supposed to emphasize good sportsmanship. Since there is no decay due to reports outside of full penalties, resetting upper tier Honors back to the baseline reassures people that the players they see with H3+ status are maintaining some semblance of sportsmanship.
There's no dishonor in being H2. It just means honor bonuses will be slow until higher tiers are reattained (which applies to everyone, so you're not missing out), and you won't get a loading screen flair. That flair is to highlight you as an honorable player, so of course it's reserved for people who've once again earned that higher status.
Since H2 is the chosen baseline for the system, there's no real reason for Riot to drop people who are below it. They don't want people pushed away from the baseline. They want their progress away from it to be due to their own persistent actions. They aren't pulling H0/1 back up to H2 at the reset because they're still on probation, but dropping them further away would be adding an extra punishment on top of the penalty they've already been given.