Wow, this is a thing? I'm so used to the European train system... wow.
If the strain stops according to the schedule, it stops. No flagging a train (not even trams). No unscheduled stops, except for emergencies. On actual high-speed routes (e.g. Germany, France, with speeds in excess of 250 kph) and with strict fire protection and evacuation rules (those up to 53 km long tunnels in Switzerland), every passenger must be seated, and unseated passengers will be offloaded before. Ticket controls have become sparse after covid-19, but are usually $100 fine + full fare + you'll end up in a database if you're get caught. And nobody pays for a reserved seat, except mostly people who really need to pop out their laptops for work. Otherwise, people are glad to sit or lie on the floor during the craziest hours (mostly Friday evenings in Germany, when students return home for the weekend).
Last I checked the schedules Amtrak had technically eliminated 'flag stops'...but, if there's no passenger listed on the manifest for an intermediate stop, and no one comes running when the train slows down, it may "stop" for about two seconds before releasing brakes and continuing on. [Edit To Clarify: This is if the train is on or behind schedule. Amtrak trains never leave before the timepoint listed in the timetable, except at a bare handful of stations designated "L" (for may Leave early). Even at a 'flag stop', if the train is ahead of schedule it will 'wait for time' at the station before proceeding.]
On Amtrak there's no additional charge for a reservation; it's built into the cost of the ticket. You're supposed to pre-purchase your ticket from the web site or telephone reservation desk if there's no ticket agent at your boarding station; while tickets can (supposedly) still be bought on board there's a stiff penalty and it will get you disliked (conductors don't like to handle cash!). Amtrak doesn't have any true high-speed routes by European standards, although the Acela Express
between Washington, D.C.; New York City; and Boston (several intermediate stops) comes close. Standard practice on Amtrak (Nota bene: I haven't traveled since pre-COVID, and the last time I took Amtrak Coach was 2012...I've been spoiled by sleeper cars!) is that you board, take a seat (at busy stations the train crew will pre-assign you to a seat, but it's done on-the-fly and not centrally by computer in advance), and then wait there until the train crew comes through to check your ticket. These days it's normally handled by scanning a QR code with the conductor's PDA, but there are still a few 'paper value tickets' which need to be physically punched and collected. At that point the conductor or his assistant will write out a 'seat check' (strip of paper about 1 inch by 3 inches with a 3-letter code for your destination) to post above your seat which marks it as taken and informs relief crews (which change every 6-8 hours) of your final destination so they know when to wake you up in the middle of the night if necessary. Once you have your seat check you are free to move about the train and visit the lounge car or cafe.
Sleeping cars on Amtrak work similarly, although sleeper space is always pre-reserved. One used to be able to upgrade from coach to unsold sleeper space on board by paying the conductor, but these days they'll tell you to call the toll-free number from your cell phone and purchase an upgrade from the central reservation desk with a credit card. One perk of sleeper space on Amtrak is that your dining car meals are inclusive with your ticket; on the Western long-distance trains which still have real kitchens and cook on board this can be very nice but on trains east of the Mississippi where they've gone to prepackaged "flex dining" much less so. Long-distance trains also have a counter-service cafe serving microwave burgers and pizza and similar fare; the nicest thing which can be said about it is that you're unlikely to starve. Although the "Track Packs" which they used to sell out of Chicago when I was new at this were worth buying; they were built around a huge fresh sweet apple with enough packaged crackers, cheese, and similar for you to nibble on for hours. Haven't seen them recently, though.
Keep in mind, also, that unlike in Europe where if you miss a train another will be along in an hour or two, outside of a few "corridors" with more frequent service Amtrak long-distance trains are 'one a day' at best...there are still a couple of routes which are tri-weekly each way, and during COVID almost the entire network outside of the corridors was reduced to every other day at best. And long-distance bus service in this country is in the process of fading away as well in many places. So if you miss, best be prepared to call an Uber...or a hotel!