How do we access these checks? In particular my goal is to monitor the health of the cluster from a single URL if possible. In the table above there is allServersAreHealthy which looks like something I'd be interested in.
You will not find them in the response of the status endpoint, or elsewhere.
If you take attention to the paragraph just above the table 10.13, you will see that this table is a list of verification (check) to do if you want to do a rolling-upgrade.
So for your use-case, you need to call the status endpoint of every node in the cluster.
Like I said it's a list of verification you have to do to make a rolling release.
Those values can be computed from the different status endpoints, and that's why this table has a Method of calculation column (to know how to compute them)