Rests get interrupted, so do inspiring moments
Some of the issue I think you are having is that you are treating the 10 minutes required time as no time.
Short and long rests occasionally get interrupted, especially in dangerous sleeping locations. If you are interrupting those, then occasionally the inspiration should as well. Roughly about 1/6th as often as a short rest does.
Interrupting rests is usually not something you need to do, but if players are abusing the rest mechanic, then an enemy patrol should have a chance to discover them. Likewise, if your party is camped outside the BBEG's room, that is probably NOT the time for an inspiring moment.
This is not limited to rests or feats, but to anything that takes time. My players once make a huge ruckus in an area where they knew a dragon hung out. Then they proceeded to continue exploring more of the town and arguing out in the open. The dragon unsurprisingly crashed their conversation. It created an interesting battle (which the characters still won) and the players learned that other creatures will react to their actions, and if given time will proceed with their own plans.
In short, the 10 minutes spent inspiring needs to be done somewhere where it is safe and makes sense to do so, hence why making it automatic is not necessarily the best idea.
Speeches aren't silent
Speaking of hostile environments, inspiring speeches are generally not delivered in whispers. Now, the feat doesn't say it has to be a speech at all, so the Rogue could just quietly go around speaking confidently to the others, in order to shore up their confidence, so this might not be a big deal.
I would however starting requiring the Rogue to tell you how they inspire the party. They shouldn't have to in-real-life have to come up with a bunch of speeches or anything, but they should say what their character is doing.
Rogue: "I give a speech about how XYZ goal is important."
DM: Are their hostiles nearby that might have heard? If so, maybe roll to see if they noticed. If not, then no problem.
Rogue: "I walk around to each party member and quietly reassure and inspire them."
DM: Are there hostiles around that might have noticed the movement or heard him moving?
Rogue: "I use inspiring leader."
DM: 10 minutes passed, did something occur?
Regardless of the method of how the inspiration is provided, things continue to happen in the world, so (occasionally) make it have consequences.
Maybe the party doesn't want their temporary hit points immediately
Generally the temporary hit points would be most useful just before a big boss fight. The cleric (or whatever) heals everyone up to full after the previous challenges, and THEN the Rogue gives their big speech. Otherwise, those temporary hit points were wasted on the little guys.
If you just give them the 8 more HP after a short or long rest, the player loses agency as to when those hit points are applied.
Consider what the Rogue gave up
By using an ASI for a feat they gave up increasing their DEX. Based on the information you provided the Rogue appears to be level 6.
Unless you roll for stats (and the Rogue got super lucky), that means their DEX modifier is only +3 (or +4 with a luck roll for stats), so they could have chosen to increase that, instead.
What do Rogues get for +1 DEX mod?
- Their to hit bonus increases by 1
- Their damage increases by 1
- Their AC increases by 1
The last point is one of the biggest: they have made themselves more vulnerable in order to help out their party.
You mention adjusting combat, but I would strongly advise against doing that. The feat is nice, especially at the current level, but it is likely only going to increase by 1 HP/level going forward, and the party members' HP gain is going to be more than 5/level in most cases. Therefore it will stop being a 20% gain for long.
So, I'm definitely with your Rogue in being adverse to any penalties or adjustments being applied to their feat. They gave up a lot for it, don't nerf it.