You talk about there being two problems here, and I agree, but I don't think those problems are incompatible. They have two different solutions, but that's to be expected. First thing, very easy: remove all national and state level speed limits on rail. Rail should be allowed to go at maximum safe design speed.
For high speed intraurban transport, we need to give a DOT-Amtrak collaboration instruction and resources, and just have them start building up lines and capacity as a massive infrastructure project.
For metropolitan transport I think that the northeastern corridor's model (state departments of transportation building and maintaining passenger rail networks that are operated by Amtrak) is probably the best option where available (interstate collaboration would be vital here). Or for a way to handle it without state cooperation, Sound Transit in Seattle is a good model. It's run by all the counties in the metro area and is building out lines of light rail as mandated by bond measures. Neither of these is a replacement for an urban center rail network (such as a subway or trolly system), but they connect towns, suburbs, districts, neighborhoods, etc. The state wide version just also connects even smaller or out of the way places.