There are legitimate cons to fare-free public transport:

• In places with large populations where fares currently make up a large portion of revenue, removing fares reduces the total funding and may necessitate cutting routes or frequency
• Many more new riders, in a system that is not receiving the automatic boost in funding that would come with fares, may result in being unable to expand to accommodate the new riders adequately
• The extra funding from tax revenue would be more efficiently spent on expanding the network (routes or frequencies) rather than further subsidising the existing routes
• Increases in “problem riders” (though the video addresses why this is overstated)

But there are many more important pros:

• Equity: it may have little impact on commutes to work, but opens up the ability for people to travel to other destinations much more easily
• Lower admin costs: no need to pay fare enforcers, no need to pay for expensive tech solutions for payment systems, etc.
• Operator safety: some studies suggest the main source of conflict between operators and passengers is at the fare box.
• Increases ridership. Duh.

The video makes a bunch more points than just this. The above is only a summary (and already quite a lossy summary) of roughly the 1st half of the video.

There are also indirect links. Transit riders are also citizens, who can lobby for better transit. At the same time transit has a lot of indirect advantages. When moving from cars to transit, there are less cars on the road, which means less lanes are needed, which then reduces cost.

As for allowing certain groups to ride free, just add it on top of existing systems or make guidelines easy. Say if you get some social security benefit, you get a transit pass as well, everybody under 18 can ride for free given they show some ID or maybe students can use their university ID to use transit.

Edit: Also free transit makes transfers easier.