The sync signal is only a sort of master-clock (derived from a master color bar or black-burst generator onboard the switcher) to time the video signals together because of varying cable lengths in big analog plants...read tv stations. Each camera, machine in those days had adjustments (subcarrier phase and horizontal phase) on each device to adjust the timing of the output video to arrive at the switcher together. If all cables are equal length, they should time up closely and all SC/HZ adjustments should be about the same.
This page is a pretty good primer on how it works;
http://www.davidstringer.info/training/timing.htm
Time code works separately from sync. The JVC machines either record TC from an internal generator (which you say you have. JVC is card based) or from a master generator (which may want sync) connected to a TC input at each machine and the edit controller. Each machine may have a selector for EXT/INT TC for the initial recording. I think your JVC's may allow re-striping TC at a later time via an edit. Playback is not affected by the switch position as the editor reads the TC output signal onlyl. I'm not sure if your JVC uses separate TC connections or if it passes through the multi-pin that JVC used in the day.
The editor reads all the TC signals and coordinates the edit based on that signal only.
Your drop-frame (which you should be using) problem is curious. I would simplify the system by eliminating some items to see if full functionality comes back. I do not know about the Alesis system so I would start there. You may have some conflicting TC running around.
As for a switcher, JVC made a simple 2RU panel in the early 80's and I used to have one. And of course, I forget the model number but if enlightenment comes to me I will add it.
It had 5 inputs, 2 outputs, luminance key input (no alpha channel), and was a full sync-generator (base-band black) as well with some obscure sync signals that only ancient guys would know. It had a simple mix bar or cut button and a downstream key button. Even some simple wipes. 1958 in a 1982 box.
I think it had multiple sync outputs for each machine if needed using the internal sync generator for each device. It also had a wonderful input 5 function. By connecting a constant on machine to input 5, it changed to a gen-lock switcher allowing that machine to become the sync generator to which all other devices on inputs 1-4 to use as a reference. In effect, gen-lock pushes the sync source upstream somewhere. That allowed you to connect video only to 1-4 and skipped the sync cables to those machines. It was great for skipping all the sync cable connections. Again, I am not sure about how the JVC multi-pin connections fit in to this concept.
Gen-lock was widely used by stations to lock to the network as a sync source so any dissolve in and out of network was properly timed and the dissolve was smooth. Not timed to net would have given you a huge horizontal jump at the end of the dissolve as you re-joined your local sync. Not to mention the SC color shift.
It got even more complicated with football remotes where the network was locked to the remote truck at some stadium somewhere. The whole chain depended on the tv truck sync. Truck>network>local station. That made switching between stadiums an adventure as the sync source changed and could be a big lurch if done live-on-air. It was usually hidden with a station break but that would be a problem for the station locked to the network while the network switched. Someones commercial usually broke up on the air.
Uncross your eyes now and keep trying and let me know how it goes,
Dave A