If you have 4x as many trains but don't load and unload on the same platform you could easily get by with 1/4th as many passengers per train.
The factors that limit train frequency are station dwell times and switching time. A subway line can generally hit 26TPH, and the top speed of most subways is usually about 70mph, with average speeds generally being in the realm of 30mph. Making trains faster actually reduces capacity; a HSR that goes 220mph is considered to have a capacity of around 4-6TPH. You can also improve throughput by cutting out all branching; Moscow gets about 40TPH as a result, which is about the feasible limit of rail systems.
30 MPH which is low = 5280 * 30 = 158,400 feet per hour / 26 TPH = 6092 feet per train. Actual subway trains are 600 feet or less long ~= 10% utilization. At 30MPH cars don't keep 126 feet between each other. Bump that to 60MPH and the trains are at 5% or less utilization and again cars don't keep 266 feet between each other even if they should. And again this is very long 600 feet subway trains most are significantly shorter than that.
Further trains have a stopping distance @ 62MPH of 820 ft with (0.15 g) deceleration. High speed trains can get an extra 0.3 m/s2 deceleration which could also be added to normal subway trains but would be an emergency situation as they knock people over.
PS: At 60MPH the theoretical limit is over 220 trains per hour assuming all trains can stop before hitting the train in front of them. But you can only approach that with full automation and multiple lines for acceleration.
We can also exclude any car stopped at a station, as you can have multiple cars unloading at the same station or have a train bypass a station without making a wider tunnel.
Sure, but we're talking about subway networks underneath major metropolises, and adding extra tunnels and platforms is basically as expensive as digging a whole new line - when it is even possible due to space or geological constraints.
Building a new network would not be constrained by existing implementations. For example, you can't just drive the trains faster on existing systems. Everything would have to be redesigned/upgraded to do it - the motors, tracks, track bed, brakes, suspension, safety equipment, schedules, signals, everything.
That doesn't answer the question. Why would the transport be "correspondingly faster" in a smaller tunnel than a bigger one, given both are new digs?
If the units/hour capacity can be met with a smaller tunnel running fast little pods on a single track, then boring out a larger tunnel that could fit perhaps three tracks would be redundant.
And that assumes that the pods could run on adjacent open tracks like railcars, whereas the intended design may require a smaller tunnel bore that is close to the size of each pod.