Hi,
I had the same problem recently and I believe I may have figured out a solution to this particular issue. Within the map layer that powers the web app you have the following settings at the bottom of the layer screen:
Once you open that up you may go further down in "Show advanced options":
Finally, I set that I want the "Start playback at" to "playback position saved with map", as shown below:
And made sure that whatever I had in that underlying map was what I wanted to be default in the web app (collapsed the ticks on top of each other).
Although this gave me quite reasonable results, I would say it was counter intuitive to have to set that in the underlying layer level as opposed to the "Time Slider" widget within the web app builder. Something for ESRI to consider in the future.