If we look in wikipedia. Here is the time in which Babar and his children’s ruled.
|Babur||1526 – 1530|
|Humayun||1530 – 1540
1555 – 1556
|Akbar||1556 – 1605|
|Jahangir||1605 – 1627|
Similarly TulsidasJi borned in year 1511. and died 1623. So above is list of Ruler that He witnessed.
Thus when Babar left the throne. Tulsidasji was just a 20 year Kid. He Wrote Ramayan in 1574. that is in the time of Akbar. Akbar was also giving due respect to him, like surdas,and Mira.
It is also said that TulsidasJi has to left ayodhya because of pandits of Ayodhya were against him because he decided to write Ramayan in prakrit language.
Now the question arise is why he has not mentioned anything about Ramas birthPlace. and not a single word for Babari Masjid. or demolition of Mandir.