Wild, thanks for sharing. And welcome to Flyertalk!
When properties manage themselves well - they likely never hear a peep most of us, surely not me. But when they fall short, I talk to them about it. Kindly, but if necessary, sternly when they aren't understanding basic CS. My DSAR then, if any exists, would then be a map for Hyatt Corporate to use to know which properties are not managing themselves properly - pointing them in the right direction of things to fix. We only become a "cautionary guest" at a single property when they can't manage the property properly in the first place.
If your DSAR has you complaining about wifi, I'm willing to bet those properties have a problem or gap in their wifi. What is really needed then is not a caution flag for other properties to know about you - but a caution flag for them to fix their darn wifi. Properties with fast wifi do not get complaints about wifi. Properties with good breakfast do not get complaints about breakfast. Properties with good fitness centers do not get complaints about fitness, etc.