UseTinyInsightsAsILogger only registers an ILogger and InsightsMessageHandler.
But InsightsMessageHandler requires IInsights so it always fails.
It might make more sense to just have UseTinyInsights() and then an additional method to add it as an ILogger implementation on the ILoggerBuilder.
Example:
.UseTinyInsights("[connection string]",
  provider =>
  {
      provider.IsTrackDependencyEnabled = true;
      provider.IsTrackEventsEnabled = true;
      provider.IsTrackErrorsEnabled = true;
      provider.IsTrackPageViewsEnabled = true;
      provider.IsTrackCrashesEnabled = true;
  })
.Logging.AddTinyInsightsLogger();