This is an issue of inadequate concourse space, not the boarding process. Restricting boarding until the determined time (by station/operations forces) has been around in the U.S. for well over a century. It started in none other than Commodore Vanderbilt's Grand Central Depot with the "Doormen" (known as "Gatemen" on the rest of the NYC and other roads, or "Usher" on others such as the PRR) controlling platform access (typically under authority of a stationmaster) for none other than the aforementioned reason of safety. In fact, the original purpose of the main concourse of GCT was for crowds to wait at the gates and the process was still used for Amtrak trains there until the end in 1991. When was the last time you saw wide open gates at an airport? Comparing a long distance/intercity railroad to a subway is comparing apples to oranges. I highly doubt this was an issue 60-70 years ago when traveling by train was much more common in this country, and concourse space was set up to properly handle the crowds. A similar problem exists at Chicago Union Station. Open up that concourse space (make it a true train "concourse") between the North and South gates and you'd see the difference. Scroll halfway down the page on this link for a historic reference, Michigan Central Station in the 1950s:
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Delays tend to exasperate the situation, but even then, with adequate concourse space the traffic flow would be a lot smoother. Unfortunately, long ago the economics of passenger railroading and terminal operation have made it seem much more worthwhile to the operators to cut the space up into retail and other uses.
I should mention that the situation seems worse in more crowded stations. Having followed the same boarding process in Seattle and New Orleans, I can tell you the experience is much better than in crowded, cluttered spaces like the Washington Union Station and Chicago Union Station boarding areas.