The Sanskrit root word for varna gives meanings like "to cover, to choose, to describe, to classify" etc.
If a blind man is named nayan-sukh, will he start seeing?
And it today's times, it certainly doesn't. Not at all.
Varna system is based on order, reciprocity, harmony etc. Any interpretation of varna system that goes against dharma is wrong.
Karma, not samsaric self-identifications are relevant as far as spiritual growth is concerned. Over-indulgence with latter cause moha, ahamkara & subsequent patan
र्व्यर्थं कुप्यसि मय्यसहिष्णुः ।
भव समचित्तः सर्वत्र त्वं
वाञ्छस्यचिराद्यदि विष्णुत्वम् ॥
In me, in you and in everything, the same Vishnu
dwells. Your anger and impatience is meaningless. If you wish to attain liberation, have equanimity.