Had a minute to come back to this after a bit more thought. Massive tl;dr alert.
I know getting our children over the hump is the focus, and a number of books and strategies are being presented as expected. But you can go a step further and ask how you can improve as a parent and/or coach. I was a coach for a bit, now happy to just be a parent.
Some of the things I have done...
I try to react to my DD's play the same as I react to any of her teammates. I don't chirp at her teammates for a mistake, so why should I chirp at her? I know we're well meaning, but I can demonstrate being more even-keel as a spectator or coach. When my kids were learning to walk, I figured out that I can't be the first to react when they fall. If I reacted, they cried 100% of the time. If I no-sold it, they only cried if they were really hurt. Sometimes they're reacting to our energy more than the event.
Evaluate how you communicate and be honest with yourself on how effective it is. I stopped talking about getting out of her comfort zone as an example. Instead I rephrased it as as growing our comfort zone. I'm not really changing the concept, but changing the phrasing can be what unlocks understanding.
As an extension of that point, take a time to revisit progress. We get so caught up in the grind that we often don't look back to appreciate how far we've come. I have a lot of video, so I can make it sport specific. But you can use school subjects as an example. There was a time when 3+5 was hard. Now we can do that with really no perceived effort.
Finally, we're big Nebraska volleyball fans in our house. If you watch enough sports together, you can see things happen in real time and bring attention to it. In the video below, you can see Ally Batenhorst make an embarrassingly bad play on a huge stage. You see Merritt Beason say "we got you" after. Then Ally goes kill, kill, block for the next 3 points. If she pouts or lets her frustration linger, does she make those plays? Our viewing of Nebraska football suggests otherwise unfortunately. LOL. But now you have tangible behavior they can model external to their day to day life. I know as a boy, we emulated our favorite baseball players growing up. It's another learning tool at your disposal. "FRS" in the title is a reference to her mentioning "Failure Recovery System" in the post game presser. She didn't detail what exactly it was that I recall but the two resources I shared in an earlier post can give suggestions on how to go about it. (I wish whoever posted the video would have combined the presser and the action.)
Exceptional wording. "Growing our comfort zone" is exactly the life lesson that everyone involved can learn from during stressful situations.