The following appears to be causing the problem on the “Desktop” layout:
In the screenshot below the “music” button is set to “Start Timeline…” the others are set to “Continue Timeline…”. The other buttons follow the same pattern e.g. the “eclectic” button has “Start Timeline…” the others are set to “Continue Timeline” (including “music”).
The net effect is on the initial opening of the layout a click of any button has all the timelines run from their beginning. However, after this first time around, when You click on another button which has its setting as “Start Timeline…” it resets itself to the start of its timeline and runs, and things work as planned.
So since having all timelines at their end (after the very first click) appears to function correctly why not do that at “Layout” opening? This seemed to be the simplest solution.
In this demo: woofsloganmenu_JHSv1.hype.zip (2.2 MB)
“On Layout Load” has the time set to “00:01.18” for all the appropriate timelines (what I believe is their end points). I only have had the time to do a slight bit of testing, perhaps there are other functions that might be affected, but things appear to work correctly.
Below: “Music” button selected with corresponding “Actions Inspector”
**Edit:** On review of the literature @jonathan already suggested this solution (as posted here) in direct reply to your question some days ago... what's up with that?