In practice whether this actually works is all over the map. Some chains will deny the claim if the cancellation terms are even slightly different, e.g. they'll say the rate on the official site was more expensive but doesn't violate the best-rate guarantee, because it allows cancellation until 6pm, instead of 5pm, which is a better set of terms. This then makes it easy to discount in third-party channels without technically violating the guarantee, by deliberately modifying the terms slightly. Other chains follow closer to the spirit of the "guarantee", but it's a bit of a lottery.

As for why they don't just offer their best rates on the official channel to begin with, without some kind of policing from the franchiser: mostly, it's market segmentation based on price sensitivity. People booking at an aggregator are more likely to have no real brand loyalty and be sorting by price, so hotel owners want to discount their rates to show up competitively in the comparison. But people booking through hyatt.com or holidayinn.com are more likely to have brand loyalty, possibly be accumulating points/status on the company dime, possibly are already committed to a specific hotel, etc. The hotel in this channel isn't competing as directly with side-by-side search results from other brands, so doesn't want to unnecessarily discount. It's in the brand's long-term interest if the official channel becomes known as having the best rates, but each individual hotel may sometimes benefit from charging more through the official channel and discounting elsewhere, hence this back-and-forth between the franchise owners and the brand management.