Do S- curve or bar chart correctly represent a progress of a project over time?

yes, 

A bar chart can represent a any type of data related to time , so we could actually find the uphills and downfalls over a time .

S-Curve and Bar chart is representation of the project progress, which depend on the feed information through milestone described in project schedule. As you follow more details i.e Level -1, 2, 3, 4...S-curve & bar chart will be more accurate.