This is more complex subject. Afrer reading different opinions I am saying varna is not by birth. But at same time there in few cases there is a chance that one may be born in particular varna.
#Hinduism
Gunas can be influenced to an extent by previous vasanas. Vasana means brain imprints.
There are hundreds of examples like this. Whole alwar saints belong to shudra or vaishya varna. But they changed varna by gaining knowledge.
Like one may have robbed someone then the man can be born in robbers family and may cause serious problems.
#HinduSaints
There is enough fluidity in varna change. Some are easy methods and some may take few years etc.
The jiva or person can be from any varna but if he understood the self and brahmin,he is greater than any jiva
To overcome these many sects started the initiation process. Ramanuja accepted everyone by giving Narayana mantra.
Chaitanya mahaprabhu intiated everyone by giving krishna mantra. Prabhupada made many shudras into great acharyas
#iskcon
He even criticized some of hinduism sects.
One of the best video on varna based on conduct and actions
#hinduism
Some cases it's birth and guna. In any case one can change varna by karma under proper guidance. He gave sacred thread to all after their performance and interest.
Alwars,nayanats, nath sampradaya Acharyas most of them belong to shudra
Some of differences are due to class than varna or caste from ithiyasa.
#hinduism
And Mahabharatam, bhagavat says especially in kaliyuga the criteria should only by charecter not even vasana
Don't form an opinion based on selective texts or videos. Finally this Chanakya episode is must watch on varna