As I said before: if you chose to ignore shitty writing - because show!Stannis is nothing like book!Stannis - then there are two possible explanations: fervent belief, and desperation. Personally, I go with the latter.
Wedding gift, yes, but she didn't gain those eggs on her own. And she hardly manipulated Khal Drogo, though my memory may be bit iffy on that detail. Burning, as you point out, was likely a fluke.
And everything else you point out she gained by possessing three dragon eggs and having people underestimate her. But had she been in the Westeros from the outset, I'd say she would have gotten eaten alive.
* she gained three dragon eggs as a gift
* her marriage to khal Drogo was negotiated by Ilyrio, who gave her those dragon eggs in the first place
* she became Drogo's favourite wife because of her unique looks and her willingness to please him - all of which was thanks to advice from her maids, which were gifted to her by Viserys
* she survived to hatch dragon eggs by accident
* she gained the Unsullied because she had dragon hatchlings and could control them
* she gained the Second Sons etc. because Daario was infatuated with her
* she gained Jorah Mormont, Ser Barristan etc. because they had literally nowhere else to go, or at least believed that to be the case
Not a single thing on this list was fully earned, all had aspects of either a gift or sheer dumb luck, even if some did have to be worked for to some degree.
* she also flew away from Mereen once things got problematic, leaving her advisors to rule
* she defeated the attack on the city because of a) dragons and b) her enemies' propensity for usage of slave armies
* she consistently opts for the most cruel or at least very cruel solution, only to have her advisors talk her out of it - her first impulse in any given situation is to burn her enemies, as well as any innocent people who happen to be in the way (she wanted to burn Quarth, Astapor, Yunkai, Mereen and King's Landing - it was always her advisors who talked her out of it; she did burn Randyll and Rickon Tarly without offering them the possibility of taking the black, or taking them as hostages)
Compare with Stannis:
* he had literally the worst starting position out of all the original contenders
** very small population base
** limited finances
** was fairly unpopular because he was not a liar and a cheat
*** everybody outside Stormlands (the area he ruled) actively hated him
* he only ever burned criminals (in the books, showrunners admittedly changed that=
* he managed to gain himself the Reach army by double-crossing his brother (murdering Renly before Renly could murder him) - albeit there he was lucky by having Melisandre
* he nearly took the King's Landing, a city usually believed to be nearly impregnable, and only failed due to sheer dumb luck on Tywin's part / Robb Stark's lack of communication skills (Lord Tully intercepted his host, allowing Tywin to be in position to help King's Landing in time)
* he defeated a Wildling army many times the size of his own host
And if you want to know the real hero of the story, Jon Snow:
* he was born a bastard, which was a disadvantage compared to other highborns (though it did allow him access to education and training)
* he forfeited his noble name to join the Night's Watch
* he was chosen as steward for Lord Commander Mormont (first time he did not earn something)
* he rose through ranks and was eventually elected Lord Commander
* he held the Wall against Wildlings until Stannis came
* he negotiated or at least helped negotiate peace with Wildlings and their inclusion into Night's Watch and, later, Stark forces
* he got himself killed by being too f***ing honorable, but got resurrected (second time he did not earn something) - this is where the books end
* he won Battle of Bastards thanks to Sansa's help (third time he did not earn something)
* he was elected King in the North
* he negotiated first access to obsidian, and then alliance with Daenerys (though there he had help from Davos Seaworth and Dany's obvious infatuation)
I could find quotes from the books, but his reasoning in nutshell was "nobody likes me anyway, so why the hell not".
ROFL, you probably spent hours typing that out and in the end Stannis and Jon Snow both were legitimate messianic figures, who legitimately believed, and were dramatically set up to be, saviors of humanity. LOLOLOL