Purpose of visualizing high dimensional data?
First of all your explanation about the methods are right. The point is that Embedding algorithms are not to only visualize but basically reducing the dimentionality to cope with two main problems in Statistical Data Analysis, namely Curse of Dimentionaliy and Low-Sample Size Problem so that they are not supposed to depict physically understood features and they are not only meaningful but also necessary for data analysis!

Actually the visualization is almost the last usage of embedding methods. Projecting high-dimensional data into a lower-dimension space helps to preserve the actual pair-wise distances (mainly Euclidean one) which get distorted in the high dimensions or capturing the most information embedded in the variance of different features.