I did a quick try on a classification problem but it seems not to respond when I try to train. The dataset is fairly small (220KB) 90000 rows x 12 columns.
As for saving the trained model, I usually save them as pickle in python but am not sure which R version does the same (maybe dput and dget?). The model shouldn't be saved on the server but the app would offer the flexibility to save it on a local directory.
Another suggestion I have is to generate tables as tables instead of text. For example, the correlation is displayed as text similar to how R generate correlation. If you could have this correlation matrix and feed it to Shiny so it output a structured table would be better.
This is definitely a good project and if the code is also open-sourced, am sure there will be more people adding more features.