If you are on a Windows computer, you can use the attached AutoHotkey script to start a song with multiple measures of count-in clicks as a workaround.
1. Start the AutoHotkey script. AutoHotkey will wait in the background for a special shortcut keystroke.
2. In Synthesia, select a song, then on the song settings screen, single click one of the mode buttons for "Watch and Listen", "Practice the Melody", or "Practice the Rhythm". (The "Song Recital" mode doesn't work with this method.)
3. Press Alt+1 to start the song. AutoHotkey will quickly send the following keystrokes:
You can press Alt+1 again anywhere in the song to restart the beginning count-in.
The "Song Recital" mode forces the song speed to 100% and you can't change it, so it doesn't work with the speed up method the script is using.
To exit AutoHotkey, right-click on its icon in the taskbar notification area and choose Exit.
I tested with current versions of AutoHotkey v1 and v2 (1.1.36.02 and 2.0.2) and with the current main and development versions of Synthesia (10.9.5903 and preview-10.9.5875). The AutoHotkey website encourages you to use v2, but if you already have AutoHotkey v1 and don't want to install v2, a v1 version of the script is included. My scripts appear to work with any of the 32-bit or 64-bit versions of AutoHotkey included in the v1 or v2 installs.