For sure, but the problem is that innumerable true situations might give rise to a given "observed" logistic curve - something that you can describe with two parameters arose from a mix of incomprehensibly complex transmission dynamics and dauntingly difficult (and often hidden) observation mechanisms. The result is that, even though after the fact we can often achieve a good fit with simple models, there's little reason to expect that simple models would have been generally predictive - the feedback loop even includes the models, since they might inform public health policy!

This isn't to say that modelling isn't useful, but that forecasting a novel infection while it's ongoing is really, really, really hard.

Edit: Also, I’d be interested to see a comparison of the fit between a typical logistic, four parameter logistic, Gompertz etc. Not all sigmoids are the same.