Because of Doctor Strange's past, Tony and the others knew a fair amount about the Sorcerer Supreme, Ancient One.
She was Earth's supreme guardian and, thanks to some hidden secret, had lived for several centuries.
Before Ancient One's death, threats from beyond Earth had always watched coldly from the sidelines, not daring to provoke her lightly.
Thanos might have been one of them. Without the Infinity Stones, no matter how incredibly powerful his body was, he could not possibly have been a match for Ancient One, the Sorcerer Supreme who commanded countless ancient spells.
Therefore, before her death, conflicts on Earth had generally remained at the level of chaotic street brawls, apart from the large-scale wars that had long since ended.
As long as those villains were not trying to destroy Earth, Ancient One would ignore them indifferently. If she had to personally handle every little incident, she would have died from exhaustion long ago.
Moreover, with control over the Time Stone, she could see countless possibilities for the future, making it even easier for Ancient One to slack off.
Unless Earth faced destruction or humanity faced extinction, she would not intervene.
Thus, Ancient One's attitude toward Leon Stark—the superhuman leader whose strength stood completely apart from the rest of the Avengers Alliance—was an extremely important basis for judgment for Tony and the others from the future.
To avoid drawing the attention of the public, or the Avengers Alliance and S.H.I.E.L.D. of this world, Tony and the others disguised themselves, then casually hailed two autonomous taxis, intending to first try their luck at the Sanctum Sanctorum in New York.
If Ancient One happened to be there, that would be ideal. If she was not at the New York Sanctum Sanctorum, they could use its portal to travel directly to Kamar-Taj.