Ok, but that temporal footprint also exists for the cross-sectional case where people try for causality with observational data, no? Whether or not we observe the earlier times, lots occurred that might confound any apparent causal relation. Just curious why growth curves generate this concern...

If I may chime in, I think this is less about some actual feature of growth curves, and more about how they are often used -- they hit a sweet spot where the descriptive understanding makes them approachable, but then people try to use them to do draw different types of substantive conclusions.

That's a common limitation of cross sectional data, yes. In general, you should not attempt to estimate a causal effect in data where you have insufficient temporal resolution. That's why a critical step in the causal pipeline is to find 'fit for purpose' data.