we used Lattice in an organization of ~150 people. We had roughly 10 leaders (exec and non-exec), responsible for various functions. The most any one leader had in their reporting tree was ~40 (one of the Eng leaders)
For Lattice, at least from what I was exposed to, the surveys ARE anonymous. They know who has and hasn't submitted a survey because everyone gets a personalized link, but that is not aggregated up into some dashboard where UUIDs are mapped back to respondents. It's something like "Bill in finance has 4 directs, and 2 of those haven't pressed submit on their unique survey." But as you can figure, that does not matter AT ALL. Anyone with a modicum of attention can figure out who wrote what when you have a reporting tree that small. You will see the responses broken out by leader, by function (say, Sales for example), and then division, whole company. The only people who had access to all the results AFAIK were the CEO, the head of IT, and HR.
In short, it's anonymity theater if your leadership has any inkling of how you communicate. I knew exactly who wrote what from my teams, and so did my leadership peers.