are all multi-center trials considered as clustered and thus they require the effect design correction in sample size calculation?
I would turn the question around: why, in the design phase of a study, wouldn't you consider all reasonable sources of variance in outcomes? Even for your 2-center example in a comment, differences in baseline outcomes or (more distressingly) in treatment effectiveness between the centers might affect study results. You want to consider those possibilities before you invest in a study, lest you spend a lot of time, effort, and money on a study that ends up ambiguously uninterpretable.
My sense is that a lot of what you might read about sample-size design has historical roots from a time before modern computers. Back then, estimates had to be based on simple assumptions about the forms of outcome distributions among individuals and centers, assumptions that allowed calculations by hand from equations and probability tables.
It's now straightforward to simulate multiple scenarios, based on pilot observations or more realistic assumptions, and see their implications for study power and sample size requirements. There are good pedagogical reasons for grounding analysts in the underlying theory exemplified in methods that can be reproduced by hand, and in simple situations those methods can work adequately. In more complex scenarios, simulations are preferable.
Quoting from Section VI.A of the 2019 FDA guidance on "Adaptive Designs for Clinical Trials of Drugs and Biologics":
Clinical trial simulations often play a critical role in planning and designing clinical trials in general and are particularly important for adaptive trials. Simulations can be used, for example, to select the number and timing of interim analyses, or to determine the appropriate critical value of a test statistic for declaring efficacy or futility. Simulations can also be useful for comparing the performance of alternative designs. A major use of simulations in adaptive trial design is to estimate trial operating characteristics and to demonstrate that these operating characteristics meet desired levels.
