Yes, that's correct. Fireball's AoE is a sphere. From the Area of Effect rules on Spheres:
A
sphere’s
point
of
origin
is
included
in
the
sphere’s
area
of
effect.
So yep, you're going to hit yourself, and yep, it's going to hurt.
As a level 1 Sorcerer, you'll probably have between 7 and 9 max hit points. The saving throw DC will probably be 13, and your Dex bonus will probably be +2 or +3, giving you about a 50/50 chance of success. If you fail your saving throw, the average damage of 28 will probably kill you instantly, under the massive damage rules. If you make your saving throw, or your race gives you fire resistance, you'll probably still be making death saving throws.
You may want to suggest to your party members that staying 20 feet away from you could be a good idea.
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
What you are missing: Bend Luck is applied on demand.
Bardic Inspiration is applied, one per turn, using your bonus action, before any dice are rolled. That means you need to decide beforehand who you think might be needing that little bump, during your own turn.
Bend Luck is applied after someone rolls a die, when you dislike the number shown. That means it takes far less thinking ahead and is far more likely to be applicable.
Especially if you take in that Inspiration is likely noticeable by a clever enemy. If they have a potent stun-spell, they might see the Bard inspiring someone and pick another target. Bend Luck still works. Likewise, the Bard might inspire the Fighter, since they seem to be in the most danger, totally missing the Assassin sneaking up on their Wizard buddy. Bend Luck might still protect the Wizard, even if the Sorcerer wasn't aware of the Assassin on their own turn.
This already gives Bend Luck and advantage over Bardic Inspiration, which will only get worse if the effect becomes more potent. A d10 on demand bonus to a saving throw means the odds of a party member failing a critical check becomes far less.
As a result, it makes Bend Luck an incredibly reliable support feature. And that then steps on the toes of the Wild Magic Sorcerer's own flavor. "Reliable" and "Wild Magic" do not belong in the same sentence.