I would be very surprised if the numbers really work out the way you claim: that the weight for carrying a passanger is similar for trains and a tube, and that the weight advantage of a tube comes essentially from 3x speed allowing for a one third the linear density of passengers. Do you have a cite?

No, this is just back-of-the envelope, admittedly. But the three factors that determine how many people you can move past a given point per unit of time are the frequency of the vehicles, the capacity of the vehicles, and the speed of the vehicles, so if you want to match the throughput of the proposed Hyperloop, but with rail traveling at 164mph, you have to either make the vehicles more frequent or higher-capacity, or both. I would expect rail to be much less frequent, at least if typical rail conventions are followed, meaning you'd need much higher-capacity vehicles, combined with the engines to bring them up to speed (which move along with the train). It doesn't take that much weight to get past what you could reasonably support on a pylon in a median... there's a reason inter-city rail isn't built this way.

> But the three factors that determine how many people you can move past a given point per unit of time are the frequency of the vehicles, the capacity of the vehicles, and the speed of the vehicles

As a matter of fact, only the frequency and the capacity matter. Speed matters to passengers but not to the amount of people you can move.

The hyperloop as proposed actually has fairly low throughput; conventional rail systems, particularly of inter-metro variety where high throughput is most important, best it easily. 28 passengers per capsule at a capsule every 30 seconds makes 3360 pph during rush hours. RER line A averages 30000 passengers per hour over the entire year, with rush hour capacity being much higher.

And if someone would like a long-distance example, at its busiest the line between Tokyo and Osaka sees up to thirteen trains per hour each capable of carrying around 1300 people.

> the three factors that determine how many people you can move past a given point per unit of time are the frequency of the vehicles, the capacity of the vehicles, and the speed of the vehicles

Why is speed relevant?

Because it's per unit of time. The faster you go, the higher throughput you'll have.

The frequency already accounts for the time component.

(People/Car)*(Car/Hour) = (People/Hour) = Throughput

At a constant frequency and higher speeds, you will simply have less trains on the track at the same time.

Yes, I guess you're right, but the maximum frequency is bounded by vehicle length and speed (you can't have a vehicle pass every thirty seconds if it takes more than thirty seconds for a single vehicle to pass a given point).

The longest TGV and Shinkansen consists are around 400 m, so they pass a given point in 30 seconds at about 50 km/h (30 mph). Of course there's spacing for safety and so on, but 50 km/h is so low that it's really a factor in practice unless you also have non-HSR traffic on the line.

A bigger factor in practical capacity is loading and offloading people and switching near termini.