Other ways to express the same idea... "The gist of my presentation is this..." "My main point is..." "The key idea, the one I hope you take home today, is..."

Yes, you can use the term "takeaway" in this context to emphasize the main point or key message that you want your audience to remember from your presentation. "Takeaway" in this context refers to the key idea, concept, or piece of information that you want your listeners to take away or remember after your presentation. By using this term, you are signaling to your audience that this is the most important point that they should remember, even if they don't remember all of the details. So, in your example, it is appropriate to say "the primary takeaway here is that they significantly improve the performance of the queries," as it highlights the main point you want your audience to remember.

It is correct. The juxtaposition "primary takeaway" sounds odd to me. It isn't wrong. However, I feel that "takeaway" is a little bit informal, while "primary" is a little bit formal. I think you are trying to maintain a somewhat informal, conversational tone in your talk. If so, then I think it would be more natural to say "the big takeaway" rather than "the primary takeaway."

primary takeaway is fine but sounds a bit robotic, most likely in a presentation you would say, “the major takeaway from this presentation is …”