I was hoping that a customizable text template was used for creating the transcript. If the transcript was created that way, it might be possible to customize the template from the Admin / Customize / Text page. Unfortunately, the use of the quote is hard coded into the plugin here:
Looking at the code, it seems that it could be difficult to use a customizable template for this. I think for now you will have to manually edit the transcripts to remove the quote.
I agree that the example you have posted with the removed quote looks better. I’ll ping @david here to see if he has any ideas about this.
I totally respect that, but there are many features that really seem to work better without quotes. Let me try to persuade you @david. For example, embedding Github gists or other Discourse topics don’t work in quotes, but if you remove the quote blocks, they get added nicely. Here is proof:
Embedding other Discourse topics for reference doesn’t work well in quotes