There are different ways in which your question can be answered. I will try to explain how existing code provisions team up to satisfy statement posted above or how the analysis steps taken by a Structural Engineer work to meet the 21.1.1 Clause. Here is the response in-terms of code clauses enforcing integrity:
1) The seismic-load-resisting system must have sufficient strength to limit ductility demand. R implies level of accepted ductility but code also imposes maximum period to be used for strength check T< CuTa. The reason: Over-estimate of period results in lower required strength. This indirectly puts a requirement for minimum strength. So code is enforcing minimum strength.
2) Similarly, the seismic-load-resisting system must have sufficient stiffness to control drift. Drift control indirectly puts a requirement for minimum stiffness. Code wants you to have minimum stiffness.
3) Critical elements are required to be designed for amplified load. (Use of Omega for collectors, elements supporting discontinuous frames etc. See AISC 341 for more detail). This is damage control.
4) The seismic-load-resisting system must have sufficient integrity to prevent separation of elements and components. This is satisfied by ensuring deformation compatibility and designing all elements for tie/ anchorage forces.
5) Stability is satisfied by requiring the seismic-load-resisting system to have sufficient strength and stiffness to ensure that second order effects are not significant. Second order check is performed. See equation below.
This ensures that even under large seismic displacements, we have sufficient ability to resist seismic load.
When doing your structural analysis to code/ ASCE 7, you are indirectly striving to achieve integrity that would satisfy much higher forces than what you have designed for. The above points just provide summary of some of the checks built in the code to ensure that.
Thanks.