Alcorn`s 8-TraXX looks fine but I am not sure if it can run the players chased (synced) each to other (in description there is written that the players are independent) that is basic requested feature for some reason. Furthermore 8 stereo tracks is way too more that I need (2 stereo or 4 mono). The same with the Binloop device that can play 32 tracks, furthermore 4U device is too big.
You seem to be in a situation that falls between the simple two channel repeaters and the more sophisticated and capable devices. Mackenzie Labs, TOA, RDL and others offer simple two channel audio players/repeaters that can be triggered by a schedule, contact closure, serial commands, MIDI commands and so on, but there seem to be few offerings that support more than two channels of audio and time code sync.
The Binloops are modular in two channel increments, you could get them with just two cards, although the frame size remains the same. However, you might want to verify if the sync requirement is really timecode chase or simply commands being sent to multiple players simultaneously. It sounds like they may actually have something like the discontinued Roland AR-3000 in mind with its AR-LINK capability to link multiple units, but that is synchronized play, stop, pause, etc. commands and not chasing to timecode.