This answer is more according to VedAnta.
"Self realisation" and "family responsibility" are Not mutually exclusive.
Both can co-exists. But that doesn't mean that both must co-exit.
A person on the path of "self realisation" starts developing indifference towards various aspects of Prakruti. "Family life" is one among them. The person doesn't have attachment towards family. That means, if he/she stays with or without family is exactly same for them.
BG 13.8-12 - ... Non-attachment and absence of fondness with regard to children, wives, homes, etc., ... Steadfastness in the knowledge of the Self, contemplation on the Goal of the knowledge of Reality --- these are spoken of as Knowledge. Ignorance is that which is other than this.
A person with indifference is neither interested towards [family] duties nor non-duties. This neither means one shouldn't perform family duties, nor it means one must perform family duties.
BG 3.17, 3.18 - But that person who rejoices only in self and is satisfied with the Self, and is contented only in the Self - for them there is no duty to perform; For them there is no purpose here at all with performing action; nor any [purpose] with nonperformance. Moreover, for them there is no dependence on any object to serve any purpose
Hence lord Krishna has described 2 approaches for self realisation:
- sannyAsa (akarma based)
- karma yoga (karma phala based)
In the 1st approach, a person explicitly leaves the family to stay away from various attachments. In 2nd approach, a person implicitly stays away from the attachments, yet being with the family.
Some people have better detachment when they stay without family and some have better detachment when they stay within family. If they try opposite to their mentality, they will likely get bewildered. How can there be only 1 right?
Since we live in a society, the Karma yoga is more applicable to larger section of people (hence better), while sannyAsa (retirement) applies to smaller section of people. Krishna equates both of them to be best in their own ways, yet Karma yoga is better due to higher relevance.
BG 5.2 — Sri BhagavAna told, SannyAsa and Karma Yoga, both are excellent doers; but between them, Karma Yoga is better than Karma SannyAsa.
Many saints have lived active family life. Yet, many yogi-s before Budhha (VyAsa, VAlmiki) and after Buddha (VivekAnanda, Ramana Maharshi) have left active family life. They are no lesser yogi-s.
Why indifference is important?
For self realisation, between "with family" and "without family", if we choose "with family".
Then after sometime
- Someone will ask us to choose between keep my spouse happy vs keep my spouse unhappy
- If spouse is happy, then someone will ask no children vs 2 children
- If 2 children, then someone will ask less bank balance vs big bank balance
- ... and this goes on
This is the reason why "indifference" is the key to self realisation. Indifference is nature of Brahman. One has to stop seeing opposites among various aspects BG 7.27.