Amtrak had and has essentially nothing to do with commuter rail outside of the Northeast Corridor, and even then most NEC commuter services are run by dedicated commuter agencies like New Jersey Transit. What Amtrak did was take over long-distance passenger service, while short-distance commuter service was taken over by state agencies. There are some similarities in the functioning and history of Amtrak and state-supported commuter rail, but they came into existence separately and remain independent of each other.

Passenger rail in general was immensely unprofitable ever since the Great Depression^1 , thanks to heavy regulation and having to compete with government-subsidized highways and airports. The railroads were deferring maintenance and making do with outdated, worn-out equipment to reduce their losses as much as possible, and discontinuing as many passenger trains as they could. Even with these deep cuts, the losses from passenger service, combined with the general decline of the rail industry in the '60s and '70s, were bankrupting the railroads. The only reason most roads continued operating any passenger service at all was that there were regulations in place that required them to get permission from the ICC before they could discontinue service. The general consensus in the industry was that long-distance passenger rail would soon be a thing of the past. Amtrak was created due to political pressure as a last-ditch attempt to save long-distance rail. Almost all the railroads were eager to get rid of their money-losing passenger trains and turned them over to Amtrak right at the beginning^2 . Because Amtrak was not required to turn a profit, they were able to invest in their infrastructure and equipment in a way that the private railroads were unable or unwilling to do. This allowed them to improve quality of service and on-time performance and led to increased ridership.

Commuter rail went through a similar process around the same time, with the main difference being that commuter rail was picked up by state (and sometimes local) agencies rather than the federal government. Many of these agencies came into existence prior to the creation of Amtrak, which meant that commuter rail losses were less of a problem for the railroads and that states already had control over commuter rail by the time Amtrak was created, which contributed to Amtrak's focus on long-distance service. Since each state approached the problem in its own way, the transition didn't happen all at once like Amtrak, and it played out in a variety of ways, though it often began with governments providing subsidies to private railroads for a while and not taking direct control until years or even decades later. For example, the MBTA began subsidizing Boston-area commuter rail in 1963 but the service was operated by the privately-owned Boston and Maine Railroad until the MBTA full took over in 1987.

Nobody can say exactly what would have happened if governments hadn't stepped in, but the fact is that most passenger rail, intercity or commuter, is not good business, and both of them were on course to largely die out if things had continued as they were prior to government interventions. Amtrak and state agencies both lose money and are totally dependent on taxpayer funding to remain operational. I don't think there's any plausible scenario in which passenger rail improves, or even survives on any significant scale, while remaining in private hands without extensive subsidies.

^1 Some specific trains were profitable because they also carried mail, but the Post Office canceled all mail-by-rail contracts in 1967. This caused a wave of service discontinuations that led to increased public pressure for the government to do something.

^2 There were a handful of railroads that initially declined to hand their passenger trains over to Amtrak, but all of those services were eventually either turned over or discontinued.