Schubert patch ideals are a class of generalized determinantal ideals. They are prime defining ideals of open patches of Schubert varieties in the type $A$ flag variety. In this paper, we adapt the linkage-theoretic approach of E. Gorla, J. Migliore, and U. Nagel to prove a conjecture of A. Yong, namely, that the essential minors of every Schubert patch ideal form a Gröbner basis. Using the same approach, we recover the result of A. Woo and A. Yong that the essential minors of a Kazhdan-Lusztig ideal form a Gröbner basis. With respect to the standard grading of assigning degree 1 to each variable, we also show that homogeneous Schubert patch ideals and homogeneous Kazhdan-Lusztig ideals (and hence, Schubert determinantal ideals) are glicci.