The GM must reveal the number on the die
Whenever a player has an ability to do something after knowing "the roll, but not the result", the GM must either tell or show the player the number on the dice, as confirmed in this pair of unofficial tweets from June 2016 by rules designer Jeremy Crawford:
Bard Combat Inspiration use with reaction to add the roll to AC "AFTER SEEING THE ROLL" So DM must show his roll to player?
If a player character ability relies on knowing a monster's roll, a DM may say what the roll is or show it
Master @JeremyECrawford, you intended that DM must communicate only the number on the dice (without added any modifiers), right?
That's correct.
Relevant modifiers or target numbers do not have to be revealed, only the raw rolls of the dice.
Therefore, it is courteous and convenient for the GM to roll the dice openly whenever there is a chance for a player to use their inspiration, ask whether the player wants to use their inspiration, and declare the hit/miss after the player replies.
A few things impact this...
You're meant to use this a lot.
Your Mantle of Inspiration is a parallel to a Lore Bard's cutting words, a Valor Bard's Combat Inspiration, and so on. This is your sub-class's 'fancy trick' with your Bardic Inspiration, and every sub-class gets a LOT of use out of their trick.
Having played a Lore Bard, I can freely say that I tended to get more use out of my Cutting Words than I did out of normal expenditures of Bardic Inspiration. The ability to reduce the total damage of AoE spells is fantastic, or make that dragon whiff when it takes a swing at my nearly-dead ally.
Burning your Reaction is not necessarily a 'low-cost,' and becomes less so as you level up.
Burning your Reaction means no opportunity attacks and it deprives several classes of special features they have (like a Battlemaster's Parry, or an Arcane Caster's Shield)
You seem to be getting an inordinately large number of Short Rests.
A Short Rest is a full hour of rest. The standard adventuring day as proscribed by the DMG is six to eight encounters, with two short rests interspersed in there. You should not have all 5 uses of your Bardic Inspiration for every combat.
Excessive Short Rests tends to screw with the balance of 5E (such as making Warlocks ridiculously powerful)
There are many things Mantle of Inspiration doesn't help against.
8 temporary HP isn't worth crap if a character gets hit with Hold Person and gets assailed with a flurry of auto-crits. But BI to boost their Saving Throw may save them from failing that save in the first place. It's better, mathematically, to successfully save vs. Fireball than to have 8 temp hp to help tank the damage. Temp HP is useless for aiding skill checks. And temp HP is only so useful if your party is having a devil of a time hitting their target.
In a typical combat encounter...8 temp HP and the chance to relocate for a Reaction is pretty powerful. But normal use of Bardic Inspiration gives your fellow players an "Oh Crap" fallback. It can let PCs succeed when they might not otherwise (such as letting the poor noisy Fighter succeed on a Stealth check).
In short...Mantle of Inspiration does precisely one thing, and does it well. This is definitely powerful in combat. But traditional Bardic Inspiration is the ultimate multitool. Need to succeed on a skill check? Need to hit with that attack? Need to succeed on a Save? Need to break a grapple before the Grell drops you off a cliff? There ya go.
Finally, Bardic Inspiration is something you can pre-load allies with. Once inspired, they can opt to use it at any time in the next 10 minutes. So if you're about to go into a boss fight, it's not a bad idea to go ahead and pass out Inspiration, leaving you clear to use your Bonus Action for other things and granting your party each one 'probably get out of trouble free' card..
Summary
If you are doing nothing but relatively straightforward combat against enemies that aren't packing Save or Suck abilities or high-damage save-based spells, versus enemies your party can reliably hit...then yes, Mantle of Inspiration will probably get used a lot. You shouldn't be able to use it as often as you do (see the bit on a your surfeit of Short Rests), but it's a good feature.
But MoI is nearly useless outside of combat and is mathematically inferior once you start dealing with most abilities that call for Saving Throws. Especially the ones that burden a character with debuffs.
Best Answer
Each creature can move up to its maximum speed
Barring speed-reducing effects (like the slow spell), there is no difference between your "current" speed and your "maximum" speed. Therefore, each creature may use a reaction to move up to its speed, no matter how much they moved during their turn. For more evidence, let's look how a single creature with multiple different speeds is handled (PHB, p. 190):
If your speed was "used up" during your turn, then the above example would work differently. You would fly 20 feet (reducing your fly speed to 40), then walk 10 feet. You would then subtract 30 (the distance already moved) from your fly speed (reduced to 40 by moving) and only be able to move 10 feet. Thus, we can conclude that speed is not "used up" during your turn.