My point was just that tracks have limits too and at some point you need to add more of them / add more train cars / add new lines.
This can be seen in Paris where RER A was built in part to decongest line 1, which then resulted in both being heavily congested (RER A is one the most heavily used commuter rail line in Europe and the whole world). This led to the creation of line 14 to decongest the A and 1 inside of Paris, but it got congested too, so now there's also the RER E to decongest the 1, 14, A and B by taking over some trips between destinations that would have required doing a change 14+A or B+A.
This made me think, how does this all compare to bikes? I found one study [1] which finds ~6500 bikes per hour for a 2m wide cycling path. That's actually much closer to a highway lane than I expected.
[1] https://fietsberaad.nl/CROWFietsberaad/media/Kennis/Bestande...
Cycling has far less density, and far greater danger. It's easy to fall off a bike, or run into someone (pedestrian or cyclist), and it's even worse if there's cars around. Bikes are also quite slow compared to trains.
The advantages bikes have are:
1. flexibility: you can go most places by bicycle if the city doesn't hinder this too much by being too car-oriented. Trains only take you between far-apart stations, but a bike can take you the door of your destination. In places with good public transit and good bicycle support, it's very common for people to ride a bike to or from the train/subway station, and use the bicycle for the "last mile" instead of walking which is much slower.
2. small size: compared to cars, bikes take little space and so they don't need huge, wide roads, or even worse, giant parking lots. Here in Tokyo, lots of places have bicycle parking lots or garages, and it's staggering how many bicycles can fit into a space that only a handful of cars would be able to park in. And in places with no designated parking, it's common for people to just park their bike on the edge of the sidewalk, in a side alley, etc., things you can't do with a car.
So back to your 6500 bikes/hour figure: the problem with this kind of comparison is that it neglects parking. If everyone is driving in a car on a highway, they need a place to park, and putting parking for cars everywhere means everything must be spaced out much, much farther, which means everyone needs a car to get around. If no one has a car and everyone has a bicycle, then you don't need any car parking and bicycle parking takes very little space, so everything is now much closer together and you don't have to go nearly as far in the first place.
Of course, this doesn't even get into quality-of-life things like lack of pollution and noise, being able to actually see people's faces as you're out and about, etc.
Even a $1000 bike would pay itself off within a year versus train fares and it is possible to buy a reasonable used commuter bicycle for hundreds less than that. Cycling is def the most affordable form of transportation out there (aside from walking of course).
Note that this is 40-60 trains per hour, and each Yamanote train holds 1724 people, giving a capacity of 69,000-100,000 passengers per hour per direction
A highway lane operating at peak capacity can carry fewer than 2000 vehicles per hour. If each vehicle held typical passenger loading of 2 passengers (average occupancy is actually lower in most cities), the yamanote line would need 17-25 highway lanes in each direction to replace one track in each direction (plus way more space for entering and exiting the highway than a subway station requires).
So while your fundamental point (train tracks have limits) is true, by that point a comparison with automobile infrastructure is truly absurd!