I hear what you are saying.
Did you read the links I posted and did you try this:
Community Hass.io Add-on: Google Assistant Webserver (broadcast messages without interrupting music).
It just might do what you are asking for as it uses google itself to ‘broadcast’ the TTS. When you trigger the ‘google broadcast function’ from say an android request, it should resume (continue in your terms) the cast after it sends the message. You can easily test if the ‘google broadcast’ interrupts your screen by asking google to do a house broadcast from another device, and see if your screen stays or goes. “hey google, broadcast…”
The problem is essentially what google speaker is controlling and what HA is controlling. If I’m playing iheart music, that is essentially google casting the audio to the speaker. If you are casting a dashboard, it is google casting the stuff (that HA is sending it) into the speaker/screen. When there is a TTS or MP3 send from the HA media player, it is HA interrupting what the Google is doing so that process ends. This [FR] and the [FR] i cited are requests to not interrupt that and possibly just mute it or something.
The link above has a good possibility of getting you where you want to be short term. We should keep asking for this to be more elegantly fixed (ala the request that was made to fix the behavior), that part doesn’t change and we should continue to lobby for that fix.
In the end a fix for one of these problems is a fix for both problems.