Yes, Jake's tone is incredibly dismissive and I think he could stand to draw it back 9 steps in this post and others, but that doesn't make his logic unsound.
For starters, the old Price routinely stopped down between showcases 1 & 2. Not always, I'm sure, but often enough. It was easier to spot in the '70s & '80s because the edit wasn't as clean. Just watch the moment Johnny says "YOUR showcase..." to the second person. The player is often staring into space unnaturally, while there's visual evidence of a cut. There's blooper reels online from Barker's last year or few where he calls for a stop after flubbing a line. Given he was the EP, if he wanted to stop, they would stop.
We talk a lot about S&P's strictness, and I don't expect many people online to know this, but an S&P presence isn't like a Brinks-level security. I think, in general, people here tend to overstate their role based on what they've read, and they overstate how complicated fixes like this are. To double check Ten Chances is not "changing a game." There's not a rubber stamp process for this sort of thing. And as I understand it, many (if not all) of the Goodson games were given more latitude than you imagine with S&P departments because they had a good reputation. If the contestant won the first prize in Ten Chances by default, nobody's mind is jumping to major foul play.
On most shows I've worked on with big money prizes and/or an outside S&P firm, most things you do elicit the S&P rep to accompany you . In all of the Price stops I've seen on tape and in person, I've seen no such person at the side of the producer or stage manager or whoever's futzing with the problem at hand.
-Jason