Level boarding is currently not possible due to the requirements imposed by the one (or two?) freight trains going through per day.
Mentioned Moscow rail has lots of freight traffic and high-level boarding.
If caltrain adopted higher platforms, they would interfere with the size of freight trains passing through.
There are ways around this: If all the high platforms were on 4-tracked sections, the freight could use the express tracks only. Mixing high and low boarding might be more painful though.
Not to mention the current train stock doesn't support level boarding, so Caltrain would be restricted to use them as express trains and only have high platforms at local stations, for example.
Or the platforms could be far away from the trains and they use extending platforms: That is potentially unreliable and costly.
There is a somewhat reasonable blog about this type of issues at http://caltrain-hsr.blogspot.com/ which is mostly reasonable though it tends to be written as if it is the only reasonable choice.
I ride Caltrain every day and the experience annoys me a lot (still less than driving), but I feel sorry for the people running Caltrain because despite providing a valuable infrastructure service that is in huge demand, they are considered the lowest priority by everyone they interact with and have to deal with the most ridiculous restrictions and regulations. For example, Caltrain knows that they are always behind schedule during rush hour and want to adjust their schedule. But to do that they have to consult everyone and their uncle over a year-long process where every nutjob's concerns about a five minute scheduling change can stall the entire process.
Infrastructure projects deal with such problems everywhere, but I have not seen a place where this attitude is so deeply engrained and systematic like here.