HI,
Sorry for taking so long to reply. It is not possible to enable time animation on a tiled layer as the tiles are static moments in time.
The reason that your time layer started speeding up is probably because your browser cache stored the results, allowing it to display faster. Below I'm going to get into the weeds of how we make requests to services, but have these three high level recommendations.
1. Reduce the amount of features in the layer
2. Reduce the complexity of the features (generalize)
3. Increase the Max Record Count (I generally don't recommend this ever, but the number of features you have is so close to a limit that this may get you by).
The Weeds:
When displaying time data in the Javascript API (what the web map is built with) time data is requested in one of two ways : Snapshot mode or On-demand. Snap Shot mode involves the web map requesting all of the features from the server in a single request and storing the data in the browser. This allows for a smooth transition from timeframe to time frame as the web map has all of the data to display up front.
On demand is triggered automatically when a request a feature layer has more data to display than the max record count set on the service. When its determined that the amount of features is to large, the web map sends individual requests to the server for each time frame. This can cause a blink or some animation issues as the web map waits for the response from the server at each time interval. Improving the server response time, like reducing the complexity of the data (vertices and/or number of fields) can reduce the lag.
The max record count on your service is 2000. You could try and increase this following these instructions to see if it helps
The reason I generally don't recommend this is that when the max record count is increased, it can bloat the amount of data being requested which can lead to a long response time from the server or cause a browser to crash. I'd suggest giving it a try and seeing if it makes a difference.
Overall, can you add an idea about this on ArcGIS Ideas as it looks like we need to improve the performance of time animation so there are no lags when using on-demand mode to request features.
-Kelly