I'm not convinced that even with finite free rides public transit users would make rush hour worse. The people taking the tube at rush hour are usually doing it where they cannot travel at an alternate, less congested time. Those people will go for free or not, and are stuck riding at the times they do. Even with a free ride card, you would still want to avoid rush hour just to not have to wait for transit to become available.

What is interesting is that most public transit obeys the same congestion rules as roads - if you take a congested road system and add more roads, you often just end up with more cars on the roads at the same times congesting the new roads again because of all the offset load that uses transit at off hours due to congestion. For people on the roads, they are effectively "free" to use once you have the car, but people still actively decide to drive at off hours to avoid congestion. The same happens with any transit system regardless of cost - unsurprisingly, people don't want to be stuck in traffic.

Brisbane, Australia already charges peak vs off-peak rates with off-peak being 80% of the peak rate. Peak hours are 6:00am-8:30am and 3:30pm-7:00pm Monday-Friday. Non-front-facing government jobs are also generally somewhat flexible in start and end times, allowing a large portion of the government's workforce to spread out around some of those times and lessen the peak demands.

In Melbourne Australia there’s a “free tram zone” where in the centre of the city anybody can use trams to get around for free. It works pretty well, although the trams sometimes get very packed.

It’s not exactly the same idea, just a similarity from where I’m living.