Why did Bhishma/Drona/Kripa join the Kauravas’ army?

It is true that Bhishma did promise to Satyavathi’s foster father that the kingdom will be ruled by the descendants of Satyavati only.

Contrary to popular perception, Bhishma did not give any promise to his dying father King Shantanu that he would protect the kingdom.

Bhishma, Drona, Kripa chose to side with Kauravas, because they being good hearted wanted to repay the debt to Dhritarasthra for the material gains they obtained. Dhritarastra himself says so.

दीर्घकालेन संसिक्तं विषमाशीविषो यथा । स मोक्ष्यति रणे तेजः पुत्रेषु मम सञ्जय ॥

महेन्द्र इव वज्रेण दानवान्देवसत्तमः । भीमसेनो गदापाणिः सूदयिष्यति मे सुतान् ॥

भीष्मो द्रोणश्च विप्रोऽयं कृपः शारद्वतस्तथा । जानन्त्येते यथैवाहं वीर्यज्ञस्तस्य धीमतः ॥

आर्यव्रतं तु जानन्तः सङ्गरान्न बिभित्सवः । सेनामुखेषु स्थास्यन्ति मामकानां नरर्षभाः ॥

बलीयः सर्वतो दिष्टं पुरुषस्य विशेषतः । पश्यन्नपि जयं तेषां न नियच्छामि यत्सुतान् ॥

ते पुराणं महेष्वासा मार्गमैन्द्रं समास्थिताः । त्यक्ष्यन्ति तुमुले प्राणान्रक्षन्तः पार्थिवं यशः ॥

यथैषां मामकास्तात तथैषां पाण्डवा अपि । पौत्रा भीष्मस्य शिष्याश्च द्रोणस्य च कृपस्य च ॥

Like a venomous snake, whose poison hath accumulated for years, Bhima will, O Sanjaya, vomit in battle the poison of his wrath upon my sons! Like the foremost of the celestials, the great Indra, smiting the Danavas with his thunderbolt, Bhimasena will, mace in hand, slay all my sons!

Bhishma, that regenerate Drona, and Kripa the son of Saradwat,--these are as much acquainted as I myself with the energy of the intelligent Bhima. Acquainted with the practice of those that are noble, and desirous of death in battle, these bulls among men will take their stand in the van of our army.

O Sanjaya, the little acceptable services that we have been able to do unto these three venerable ones, will certainly be repaid by them owing to their own noble dispositions.

——

The statements of Bhishma, Drona and Kripa, when Yudhisthira approached them before start of the war, confirm that they were indebted to Kauravas and thus, they were acting like eunuchs due to that reason alone.

I have been bound by the Kauravas with (their) wealth. It is for this, O son of Kuru's race, that like a eunuch I am uttering these words.