It's probably the case that someone, somewhere has tried this at one time or another. I'm not aware of any mass marketed camera that has done so. But there are many, many cameras that I've never studied in this regard.
I'm also wondering how long the "reset" takes on typical curtains: Does reset contribute to the maximum frame rate of cameras?
Not usually.
Most modern DSLRs, which pretty much universally have electronically controlled vertical focal plane shutters, have curtain transit times in the neighborhood of 3-5 milliseconds. This gives them flash sync speeds of around 1/250 to 1/160 seconds (4-6 milliseconds). Keep in mind that at flash sync speed, in addition to the first curtain being fully open before the second curtain begins to close, the shutter must also remain fully open long enough for a flash to discharge most of its energy before the second curtain begins to close. Most shoe mounted speedlights only need 1-2 milliseconds to do a full power dump. Studio strobes tend to take longer, which is why one must sometimes use a shutter speed slower than the camera's X-sync when working with such studio lights.
The curtains for larger-sensored FF cameras obviously need to travel further than the curtains in front of cropped sensors, but those cameras also tend to have faster, more robust shutter assemblies with faster transit speeds which allow roughly the same total transit time over the longer distance.
The fastest current DSLRs with shutter curtains resetting and the mirrors cycling between each frame can burst at 12 (Nikon D5) to 14 (Canon 1D X mark II) fps. For those same cameras to do 14 (D5) and 16 (1D X Mark II) fps they need to leave the mirror locked up in burst mode. One fourteenth of a second is 71 milliseconds. This 71 milliseconds would include time for:
[mirror moves up→mirror up confirmed→shutter "fire" command sent→shutter curtains transit the sensor to expose the picture→shutter confirmed closed→release command sent to mirror→mirror down/shutter curtain reset→AF/metering→repeat]
Some of that 71 milliseconds is occupied by the exposure time of about 4-5ms if the shutter time is shorter than the flash sync speed. Some of that is taken up by AF and metering. Half of that 71 milliseconds less the 5ms needed to actually take the picture and 15-20 milliseconds needed to AF/meter is about 23-25 milliseconds each way for the mirror movement. So the limiting factor is the time required to move the mirror, as evidenced by the two cameras' faster burst rates when the mirror is locked up for a complete burst.
This means that in theory the shutter curtains have plenty of time to spare to reset while the mirror is dropping back down to allow the camera to AF and meter before beginning the cycle again. But that is not the way most DSLRs do it. For whatever reason¹, most of them wait until just before the mirror is completely down to reset the shutter curtains. They usually begin resetting with just enough time to complete the reset by the time the mirror is all the way back down and at rest.
This popular youTube video shows a Canon 7D shutter assembly at various high shutter speeds. As you can plainly see, the shutter curtains rest in the bottom position after the exposure is captured until the mirror drops almost all the way back down. Only then do they start to move up to reset for the next shot. Yet they are still completely reset by the time the mirror stops bouncing around at the end of its movement.
You can also see that the mirror movement time, both up and down, as well as the pause needed to confirm the mirror up position and send the signal to the shutter to fire, is several times longer than the time the shutter curtains need to transit the sensor. The video is running at 1/400th actual speed and the entire cycle when four are shown at once takes 31 seconds, of which only slightly less than two of those seconds is required for the shutter curtains to transit the sensor when the exposure is captured. That translates to just less than 5 milliseconds out of the 77.5 millisecond total. The Canon EOS 7D has a maximum burst rate of 10 fps, which equates to 100 milliseconds per cycle. That allows a tad over 20 milliseconds for the AF and metering systems to do their thing before the mirror starts back up for the next frame.
¹ The delay in beginning the shutter curtain reset is probably to insure that the sensor is read out completely before the shutter curtain begins to reset. Even though they might should stay together and thus let no light past during the reset, waiting until the sensor is read allows less than perfect precision in this respect. Or perhaps there is less stress/wear on the shutter curtains and the assemblies that drive each one if they are allowed to be slightly separated as they reset?
I am wondering if any (or most, or even all) curtain shutter cameras, past or present, don't "reset" the curtain between shots, but rather let it run from whichever post it last wound back to the other post on the next shot?
I am aware of a few cameras that have done something similar in a "halfway" manner for certain "silent shooting" modes in Live View.
Silent shooting modes affects the way some Canon EOS DSLRs with "silent shooting" options in Live View cycles the mirror and shutter curtain for each shot you take.
The following is specific for the EOS 5D Mark III introduced in 2012. Other EOS DSLRs with 'Silent Shooting' modes in Live View from the same time period, such as the 5D Mark II (2008) and the 7D (2009), are similar. The 7D Mark II (2014) is different.
First off, the mirror stays up as long as you are in Live View and are using Contrast Detection Autofocus or Manual Focus.
Mode 1 begins with the first curtain open prior to the shot (so that the sensor is exposed to give a Live View image on the camera's LCD screen). When the shutter button is pressed the sensor is cleared and then electronically exposed from bottom to top (top to bottom of the inverted image) to begin capturing the image. The exposure is ended by the movement of the first curtain from the bottom of the light box back to the top of the light box. This is the same direction the first curtain is normally moved when it is reset following a still shot taken using the viewfinder. The second curtain never moves throughout the entire Live View - Silent Shooting cycle with mode 1. For each frame exposed, the only sound is the first curtain closing to end the exposure and then reopening for the next shot (if the shutter button is held down for a continuous burst) or to allow the sensor to send an image to the LCD screen for Live View (if the shutter button is released).
Mode 2 is a single shot mode. The shot begins just like Mode 1. But the second curtain closes from the top to the bottom of the light box (bottom to top of the inverted image) to end the exposure. Once the second curtain closes it doesn't reopen to expose the sensor for Live View until the full press of the shutter button is released. The first curtain never moves throughout the entire Live View - Silent Shooting cycle with mode 2. For each frame exposed, the only sound is the second curtain closing to end the exposure. Once the full press of the shutter button is ended the sound of the second curtain reopening to allow the sensor to send an image to the LCD screen for Live View will be heard.
Disable allows shooting with Canon's Tilt-Shift lenses or when using extension tubes (and presumably any third party lens that has an aperture manually set on the lens). When the shutter button is fully pressed using this setting the first curtain resets to the top of the light box, the sensor is cleared, and the first curtain reopens to begin the exposure. The exposure is ended by the conventional movement of the second curtain from the top to bottom of the light box, the sensor is read out, and the second curtain is reset to the top of the light box while the first curtain remains open in the bottom of the light box.
