I mean, how crowded are these trains, especially if we've sprung for
first class STPs? This would be a weekday mid-April, and we're willing
to get started first thing in the morning. Is it really a gamble? What
happens if you don't have a seat reservation and the car is full? Do
you stand? Are you kicked off?
I have taken this route often without any reservations (2nd class). Never had to stand. People get on and off all the time - locals, people out for a day hike, etc. - seats open up at every stop. Take one before the people entering at that station get on the train.
No, you will not be kicked off. They can pile a lot of people into a train! Go to the first class car that is furthest away from where most people are getting on, for some reason people don't like walking any distance along the platform for a less busy car, or once on the train ask the conductor where it is less busy, if people are taking seats up with their backpack just ask “is this seat free” and they will move their stuff for you. Lots of times people won't make room unless you ask them to.
Conversely, paying for a reservation is just a fraction of the cost of a holiday, so it’s also not a big thing, in my opinion.
Since you are interested in history, the Belle Epoque might be fun for you. I have only taken it once, but it is a unique train.