It's possible with rail. You just need to avoid stopping trains on the main line. Switch off on to a siding to serve the station. Switching should be at the fastest safe speed; slow trains do not belong on the main line.
a. You show up, only to discover that the train just left. Since it isn't properly scheduled, you just have to get lucky. Enjoy waiting an hour or two for the next train.
b. You show up, only to discover that there are too many people to fit on the train. Since you don't have a seat reservation, the train fills up and you don't get to board. Enjoy waiting an hour or two for the next train.
Train travel is much more pleasant than air travel and being able to just turn up and go, and carry your own luggage, is a big part of that. This goes beyond fixing what's not broken - why are you trying to make the pleasant thing like the less pleasant thing?
Reality: that won't happen
It costs money to run trains that might be mostly empty, so they are run less often, which makes them less used, which makes them run less often, and so on.
At best, reality looks more like this: https://www.mbta.com/schedules/CR-Lowell/timetable
That's 21 trains per weekday. Weekend days get only 8 trains.
Amtrak is much much worse. At that service level, you could get 1 train per week, seasonal.
Reliable trains are a good thing. Frequent trains are a good thing. Your proposal doesn't help with either of those things and would waste money that could be spend improving them instead.