What your argument boils down to is that you have to go and murder something to be allowed to rest again. No other activity can be used to separate short rests.
And... enjoy your game, it's neither how the rules are written, how they're intended, or makes any sense at all.
"You have a limited well of stamina that you can draw on to protect yourself from harm."
When I have an ambiguous rule to adjudicate at my table, I look at how it works within the story. In this case, I view it from its description as when the hero in a movie gets knocked around a bit and is starting to look punch drunk, but he manages to shake it off and fight with renewed vigor. I don't really view it as a means of repeatedly shaking off the weariness of battle. Healing completely just by "shaking it off" or "just feeling better" after a couple of hours does not seem to make sense, but getting a good night's rest might work.
Mechanically, I would have preferred that the feature was a reaction that gave temporary hp after taking damage as long as you were still able to take actions (not incapacitated, unconscious, etc.). But that is not how it was written, and I am not big on house rules. I choose to find story narrative reasons for why the rules are what they are. For the most part, the rules also work as a cohesive unit. If you start pulling on one part, you are going to have to shore up another.
Beyond that, it really seems like this argument is a solution in search of a problem. Just spend some hit dice, drink a couple potions or hit up your healer for healing and move on. Part of encounter/adventure design is getting the party to spend resources, and hit dice are one of those resources. If there is no spending of resources, the party (specifically in this case the fighter) is not really stretched. Caution can be thrown to the wind because the fighter can just spend a quarter of a day after every fight convalescing.
When this came up in the past, Mike Mearls has said the design was for there to be a short rest of 1 to more hours in length. He also followed that by saying that people could agree to play it as a series of 1 hour rests chained together if they really wanted to do so.
As others have said, please re-read the forum rules that you agreed to when signing up here. We are not ridiculous for holding an opinion that fits well with how the rules are written mechanically as well as fits with the story description of the rule as well as being how the feature was intended to work.