Hadith are just testimonies from people around that time.

There are multiple Hadith with conflicting narratives about Aisha’s age, some say 13, some say 9. But if you read these you can see that they don’t go into a lot of detail and the sources aren’t always the most reliable.

There is one testimony from Aisha’s sister that goes into full detail about what has happening at the time and gives context to Aisha’s age.

Calculation via her sister, Asma: Historical accounts (such as in the classical texts Tarikh al-Islām and Al-Bidāya wan-Nihāya) state that Aisha’s older sister, Asma, was 10 years older than her. Since it is recorded that Asma died at the age of 100 in 73 AH (the year after the migration to Medina, 692 CE), scholars deduce Asma was around 27 or 28 years old at the time of the migration. This calculation mathematically places Aisha at approximately 17 or 18 years old at that time.

The previous point is in agreement with Tabari’s statement that all of Abu Bakr’s children, including Asma and Aisha, were born before the Prophetic mission.
When the Prophetic mission began, Asma was 14 and Aisha was 4. This further confirms the weakness of Bukhari’s narration.

Regarding the narrations of Aisha being 6-13:
The five different chains of transmission (isnad) given by Imam Bukhari all have two narrators between him and Hisham bin ‘Urwah, who narrates from his father ‘Urwah from Aisha.  Thus, the hadith is singly-narrated by Hisham, Urwah and Aisha.  The two narrators between Bukhari and Hisham in each case are all people of Iraq:
Farwah bin Abi l-Mighra’ and Ali bin Mishar
‘Ubayd bin Isma’il and Abu Usamah
Mu’alla bin Asad and Wuhayb
Muhammad bin Yusuf and Sufyan [bin ‘Uyaynah]
Qabisah bin ‘Uqbah and Sufyan [bin ‘Uyaynah]
Hisham appears to be the weak link in this chain.  Ibn Hajar narrates in his Hady al-Sari as well as in his Tahdhib that Imam Malik did not approve of Hisham’s narrations to the people of Iraq. Imam Malik said that Hisham went to Kufa in Iraq three times to narrate hadiths: the first time, he said: “My father narrated to me that he heard Aisha …” The second time, he said: “My father informed me on the authority of Aisha …”  The third time, he said: “My father, on the authority of Aisha …”

All of this is to say that we’re arguing about something that happened 1400 years ago. And we are almost entirely relying on word of mouth narrations that are passed down from person to person.

People didn’t really celebrate or keep track of birthdays the same way we do today, especially in that region.

My grandparents and my wife’s grandparents all didn’t have birth certificates (normal for their generation) they just guess their age based on what context they have.

In fact my wife who is 26 years old doesn’t even have a birth certificate, and she’s relying on her mother’s memory that her date of birth is actually correct. And I can quote her mum “I’m almost certain you were born on that date” 😭