You
don't have to keep it open. Or at least I've never had to. I just tried it, in fact, without issue.
Perhaps you simply misread. From the article you posted:
"When the iPad connects to the iPhone, a blue bar appears at the top of the iPhone's screen. This indicates that a device is connected to the Personal Hotspot. The iPad can access the Internet via the iPhone as long as the Personal Hotspot is turned on and the iPad is in Wi-Fi range of the iPhone.
You can use the iPhone as you normally would even while the iPad is tethered to it. Personal Hotspot doesn't interfere with it. The only difference you may notice is that the iPhone's Internet connection may be a little slower than normal since it's being shared with the iPad."
Emphasis mine.
https://support.apple.com/en-us/HT204023