The Chinese President has literally sent a message
I think too much credit is given to Kissinger or any other individual. Kissinger was able to do what he did because the strategic situation made it in the interest of both America and China to align against a common foe. In order for there to be another "Kissinger" (e.g. a messenger), that messenger will need to bring an offer that fits the needs and imperatives of both parties.
What that offer is at this point is uncertain, but given that the greatest threat both China and US perceive is each other (at least at this point), I, unfortunately, think there can be no "Kissinger moment"
Conditions would have to change where one side and/or the other truly consider the other side not a threat in any shape or form, or some existential problem greater than the opposing side <mandates> cooperation. There's been talk that COVID-19 or global warming or terrorism could be that unifying enemy, but I think it is highly doubtful any of those is existentially threatening enough and requires cooperation to give that opening.
Maybe the best we can hope for is John Kerry making a limited climate deal between the US and China, but no one will mistake it for a Kissinger moment.