Not sure if this is an option for you, but you could set your ASP.NET Core app to target the .NET Framework.
Edit yourapp.csproj and change:
<TargetFramework>netcoreapp2.2</TargetFramework>
to
<TargetFramework>net472</TargetFramework>
Now you should be able to add the reference to System.Speech and do something like:
System.Speech.Synthesis.SpeechSynthesizer synth = new System.Speech.Synthesis.SpeechSynthesizer();
synth.SetOutputToDefaultAudioDevice();
synth.Speak("Your awesome web site is starting!!");
Another cool way to get this done would be to use some on-line API, such as Azure Speech Services (text-to-speech) that is pretty much platform independent and provides plenty of configuration options (https://learn.microsoft.com/en-us/azure/cognitive-services/speech-service/text-to-speech).