feat: Improve live latency on load (#5268)

With this change the latency is reduced in 50ms approx. Very important
for LL streams.

lib/media/playhead.js

@@ -251,7 +251,7 @@ shaka.media.MediaSourcePlayhead = class {
     this.videoWrapper_ = new shaka.media.VideoWrapper(
         mediaElement,
         () => this.onSeeking_(),
-        this.getStartTime_(startTime));
+        () => this.getStartTime_(startTime));
 
     /** @type {shaka.util.Timer} */
     this.checkWindowTimer_ = new shaka.util.Timer(() => {

lib/media/video_wrapper.js

@@ -27,17 +27,25 @@ shaka.media.VideoWrapper = class {
   /**
    * @param {!HTMLMediaElement} video
    * @param {function()} onSeek Called when the video seeks.
-   * @param {number} startTime The time to start at.
+   * @param {function():number} getStartTime Calle to get the time to start at.
    */
-  constructor(video, onSeek, startTime) {
+  constructor(video, onSeek, getStartTime) {
     /** @private {HTMLMediaElement} */
     this.video_ = video;
 
     /** @private {function()} */
     this.onSeek_ = onSeek;
 
-    /** @private {number} */
-    this.startTime_ = startTime;
+    /** @private {?number} */
+    this.startTime_ = null;
+
+    /** @private {function():number} */
+    this.getStartTime_ = () => {
+      if (this.startTime_ == null) {
+        this.startTime_ = getStartTime();
+      }
+      return this.startTime_;
+    };
 
     /** @private {boolean} */
     this.started_ = false;
@@ -58,7 +66,7 @@ shaka.media.VideoWrapper = class {
         HTMLMediaElement.HAVE_METADATA,
         this.eventManager_,
         () => {
-          this.setStartTime_(this.startTime_);
+          this.setStartTime_(this.getStartTime_());
         });
   }
 
@@ -86,7 +94,7 @@ shaka.media.VideoWrapper = class {
    * @return {number}
    */
   getTime() {
-    return this.started_ ? this.video_.currentTime : this.startTime_;
+    return this.started_ ? this.video_.currentTime : this.getStartTime_();
   }
 
 
@@ -103,7 +111,7 @@ shaka.media.VideoWrapper = class {
           HTMLMediaElement.HAVE_METADATA,
           this.eventManager_,
           () => {
-            this.setStartTime_(this.startTime_);
+            this.setStartTime_(this.getStartTime_());
           });
     }
   }